At the end of sigma ( phi ( ) ) : From "5" to "5 figures" is a reference to the existence of additional even solutions to the equation :
phi ( sigma ( n ) ) = sigma ( phi ( n ) )
Some small ones are exhibited here .
In Section 1 are dozens of specific expressions which generate
multitudes of solutions .
In Section 2 is an update on the computation reported in
sigma ( phi ( ) ) : From "5" to "5 figures" and 2 new
not-so-specific expressions which generate thousands of new solutions .
Count = 3121
The lists of examples below are all complete for all necessary sigmas
less than 2^53 .
For prime p > 5, if (p-1)/2 is prime, and if p+1 has no prime factors other than 2, 3, and 71, then 2 * 3 * 3833 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 8747, 61343, 995327, 38654387, 609880343, 20352466943, 146703225167. For prime p > 5 and != 47, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 37 and 61, then 2 * 3 * 47^2 * p is a solution. E.g., p = 7, 11, 23, 107, 383, 863, 887, 8747, 8783, 14207, 35963, 89303, 511487, 591407, 632447, 995327, 2248703, 2913083, 5447543, 26433983, 52867967, 59476463, 111233987, 158603903, 196111583, 465567743, 485720063, 577214207, 931135487, 1619275103, 2636655947, 3549954047, 3896195903, 4474497023, 4951415627, 6147553403, 7792391807, 38758514687, 55412563967, 69844101983, 273649201523. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, and 151, then 2 * 3 * 43487 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 3623, 7247, 8747, 587087, 995327, 44325143, 150294527, 315201023, 1202356223, 1322093183, 1711948607, 2644186367. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, and 877, then 2^2 * 3 * 36833 * p is a solution. E.g., p = 7, 11, 23, 47, 83, 107, 167, 383, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 40823, 81647, 217727, 244943, 263423, 338687, 489887, 774143, 995327, 1210103, 2667167, 4148927, 10229327, 10450943, 14751743, 27433727, 29647547, 48009023, 74680703, 88942643, 96018047, 98018423, 107407943, 119042783, 125411327, 218225663, 334158047, 516854687, 2425739903, 3437054207, 3491610623, 3528663263, 3866685983, 4116773807, 4514807807, 4980788063, 6367567247, 6983221247, 8538493823. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, and 307, then 2^3 * 3 * 9209 * p is a solution. E.g., p = 7, 11, 23, 47, 59, 107, 179, 359, 383, 479, 719, 863, 1439, 1619, 2879, 2999, 4799, 5399, 8747, 10799, 21599, 23039, 25919, 40499, 51839, 67499, 71999, 134999, 138239, 143999, 165779, 233279, 265247, 349919, 368399, 691199, 995327, 1180979, 1199999, 1474559, 2303999, 2361959, 2486699, 2652479, 2915999, 2949119, 2984039, 3110399, 3732479, 4976639, 5467499, 5654939, 5658623, 6560999, 7464959, 9047903, 9548927, 11317247, 12287999, 14929919, 29859839, 33929639, 36863999, 39321599, 42439679, 44236799, 59999999, 78643199, 85030559, 111901499, 128910527, 223948799, 255091679, 311039999, 379687499, 471551999, 491519999, 566870399, 589823999, 596807999, 622079999, 759374999, 863999999, 943103999, 1133740799, 1179647999, 1193615999, 1349999999, 1399679999, 1530550079, 2267481599, 2590312499, 2685635999, 3627970559, 5308415999, 5371271999, 5739562799, 7644119039, 9431039999, 9492187499, 12375410687, 13814999999, 14062499999. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 37, and 211, then 2^2 * 3 * 93683 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 887, 8747, 14207, 22787, 35963, 187367, 511487, 591407, 995327, 2913083, 34192127, 107923967, 465567743, 746717183, 931135487, 1619275103, 4376592383, 4474497023. For prime p > 5 and != 11 and != 23, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 31, and 103, then 2 * 3 * 5^2 * 617 * p is a solution. E.g., p = 7, 47, 107, 383, 863, 1487, 8747, 9887, 29663, 79103, 103787, 344843, 369023, 995327, 2452223, 3736367, 13284863, 24828767, 26569727, 29890943, 48758783, 102518783, 157352687, 331395083, 521453567, 615112703, 658243583, 1230225407, 1316487167, 9909782027, 11664359423. For prime p > 5 and != 11 and != 47 and != 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 127, 157 and 271, then 2^6 * 3 * 941 * 4877 * p is a solution. E.g., p = 7, 23, 383, 863, 1523, 8747, 717803, 995327, 1435607, 2366303, 3525167. For prime p > 5 and != 7 and != 47 and != 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 271, and 2551, then 2 * 3 * 5 * 4877 * 20407 * p is a solution. E.g., p = 11, 23, 383, 863, 8747, 995327. For p = 7 or prime p >= 863, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 17, 103, and 271, then 2 * 3 * 5 * 4877 * 21011 * p is a solution. E.g., p = 7, 863, 2447, 3467, 5507, 8747, 9887, 19583, 29663, 42023, 52223, 55487, 79103, 84047, 124847, 235007, 313343, 626687, 995327. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, and 8821, then 2 * 3 * 158777 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 8747, 995327, 34296047, 617328863. For p = 23 or 83 or prime p >= 167, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 157, and 271, then 2^2 * 3 * 5 * 13 * 941 * 4877 * p is a solution. E.g., p = 23, 83, 167, 383, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 40823, 81647, 211007, 217727, 244943, 263423, 338687, 364223, 369263, 489887, 774143. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 109 and 251, then 2 * 3 * 984923 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 1307, 8747, 10463, 27107, 995327, 1512023, 3814127, 10265183, 18505727, 22884767, 46193327, 148045823, 164242943, 578617343. For p = 7 or prime p >= 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, 31, and 1531, then 2 * 3 * 5^2 * 119417 * p is a solution. E.g., p = 7, 107, 383, 467, 863, 1487, 2027, 8423, 8747, 16223, 103787, 369023, 619007, 995327, 1314143, 1819583, 2976263, 3736367, 7195967, 13284863, 18225023, 18629207, 18690047, 21835007, 24564383, 26569727, 29890943, 30553847, 37179167, 40738463, 48758783, 106445663, 128753663, 129527423, 156918527. For prime p > 5 and != 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 31, and 307, then 2 * 3 * 5^2 * 9209 * p is a solution. E.g., p = 7, 23, 47, 59, 107, 179, 359, 383, 479, 719, 863, 1439, 1487, 1619, 2879, 2999, 4799, 5399, 8747, 10799, 21599, 23039, 25919, 29759, 40499, 51839, 66959, 67499, 71999, 103787, 111599, 133919, 134999, 138239, 143999, 150659, 165779, 233279, 238079, 265247, 301319, 349919, 368399, 369023, 464999, 602639, 691199, 995327, 1071359, 1142039, 1180979, 1199999, 1474559, 2142719, 2303999, 2361959, 2486699, 2652479, 2678399, 2915999, 2949119, 2984039, 3110399, 3571199, 3719999, 3732479, 3736367, 4976639, 5467499, 5654939, 5658623, 6560999, 6852239, 7464959, 9047903, 9136319, 9548927, 11317247, 11427839, 12287999, 13284863, 13704479, 14929919, 26569727, 27408959, 28569599, 29062499, 29859839, 29890943, 33929639, 34283519, 36863999, 39321599, 42439679, 44236799, 48758783, 58124999, 59999999, 60263999, 60948479, 68567039, 71377499, 74399999, 78643199, 83699999, 85030559, 97627679, 107135999, 110822519, 111901499, 120527999, 128910527, 131563007, 135593999, 137134079, 214271999, 223432499, 223948799, 228556799, 255091679, 311039999, 379687499, 471551999, 491519999, 508477499, 566870399, 589823999, 596807999, 622079999, 622727999, 658243583, 720749999, 759374999, 863999999, 943103999, 1133740799, 1179647999, 1193615999, 1284794999, 1316487167, 1349999999, 1388482559, 1399679999, 1506599999, 1530550079, 1729799999, 1911774959, 2267481599, 2590312499. For prime p > 5 and != 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 7, and 3067, then 2^2 * 3 * 5 * 92009 * p is a solution. E.g., p = 7, 11, 23, 47, 59, 83, 167, 179, 359, 383, 479, 503, 587, 719, 839, 863, 1439, 1619, 2099, 2879, 2999, 3023, 3779, 4703, 4799, 5399, 5879, 6047, 6719, 7559, 8747, 8819, 10079, 10799, 12347, 14699, 21599, 23039, 25919, 26459, 29399, 37799, 40499, 40823, 51839, 52919, 53759, 67499, 71999, 81647, 134999, 138239, 143999, 158759, 170099, 188159, 201599, 217727, 233279, 244943, 257627, 263423, 264599, 338687, 349919, 377999, 453599, 470399, 483839, 489887, 510299, 514499, 587999, 671999, 691199, 774143, 839999, 907199, 952559, 995327, 1008419, 1020599, 1028999, 1049999, 1104119, 1180979, 1199999, 1210103, 1270079, 1474559, 1505279, 2041199, 2057999, 2303999, 2361959, 2551499, 2667167, 2778299, 2915999, 2949119, 3110399, 3732479, 3937499, 4148927, 4976639, 5375999, 5467499, 6482699, 6560999, 7464959, 7654499, 7717499, 9525599, 10321919, 10450943, 11593259, 11777279, 12287999, 13778099, 14117879, 14405999, 14751743, 14929919, 15308999, 15434999, 16669799, 17860499, 18007499, 18370799, 22049999, 23186519, 26459999, 27433727, 29393279, 29647547, 29859839, 31116959, 33339599, 34836479, 35720999, 36741599, 36863999, 39321599, 44099999, 44236799, 47416319, 48009023, 49601159, 51525599, 53343359, 59999999, 63787499, 66679199, 69672959, 71441999, 74528099, 74680703, 78643199, 82319999, 83348999, 85030559, 88942643, 90016919, 92009999, 96018047, 97383383, 98018423, 112877867, 112895999, 115762499, 119042783, 120022559, 125411327, 151262999, 151485263, 154828799, 165957119, 167999999, 173604059, 176732807, 192675839, 208372499, 223948799, 243458459, 252104999, 255091679, 270509399, 301055999, 302970527, 309657599, 311039999, 314999999, 329763839, 335999999, 379687499, 385351679, 423359999, 430079999, 445181183, 486916919, 491519999, 515255999, 541018799, 566870399. For prime p > 7 and != 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, and 3067, then 2 * 3 * 5 * 7 * 92009 * p is a solution. E.g., p = 11, 23, 47, 59, 179, 359, 383, 479, 719, 863, 1439, 1619, 2879, 2999, 4799, 5399, 8747, 10799, 21599, 23039, 25919, 40499, 51839, 67499, 71999, 134999, 138239, 143999, 233279, 349919, 691199, 995327, 1104119, 1180979, 1199999, 1474559, 2303999, 2361959, 2915999, 2949119, 3110399, 3732479, 4976639, 5467499, 6560999, 7464959, 11777279, 12287999, 14929919, 29859839, 36863999, 39321599, 44236799, 59999999, 74528099, 78643199, 85030559, 92009999, 112877867. For prime p > 5 and != 11 and != 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, and 3067, then 2 * 3 * 5 * 11 * 92009 * p is a solution. E.g., p = 7, 23, 47, 59, 179, 359, 383, 479, 719, 863, 1439, 1619, 2879, 2999, 4799, 5399, 8747, 10799, 21599, 23039, 25919, 40499, 51839, 67499, 71999, 134999, 138239, 143999, 233279, 349919, 691199, 995327, 1104119, 1180979, 1199999, 1474559, 2303999, 2361959, 2915999, 2949119, 3110399, 3732479, 4976639, 5467499, 6560999, 7464959, 11777279, 12287999, 14929919, 29859839, 36863999, 39321599, 44236799, 59999999, 74528099, 78643199, 85030559, 92009999, 112877867. For prime p > 5 and != 383, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, and 5101, then 2 * 3 * 3305447 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 863, 8747, 995327, 1469087. For p = 11 or prime p >= 59, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 7, and 79, then 2^3 * 3 * 5 * 13 * 23^2 * 29 * p is a solution. E.g., p = 11, 59, 83, 107, 167, 179, 359, 383, 479, 503, 587, 719, 839, 863, 1439, 1619, 2099, 2879, 2999, 3023, 3779, 4703, 4799, 5399, 5879, 6047, 6719, 7559, 8747, 8819, 10079, 10799, 12347, 14699, 18959, 21599, 23039, 25919, 26459, 29399, 37799, 40499, 40823, 45503, 51839, 52919, 53759, 66359, 67499, 71999, 81647, 119447, 134999, 138239, 143999, 158759, 170099, 188159, 201599, 217727, 227519, 233279, 244943, 263423, 264599, 338687, 349919, 377999, 379199, 453599, 470399, 483839, 489887, 510299, 514499, 524243, 587999, 650327, 671999, 691199, 774143, 839999, 907199, 952559, 995327, 1008419, 1020599, 1028999, 1049999, 1180979, 1199999, 1210103, 1254203, 1270079, 1327199, 1474559, 1505279, 2041199, 2057999, 2303999, 2361959, 2551499, 2667167, 2778299, 2787119, 2915999, 2949119, 3110399, 3145463, 3185279, 3344543, 3732479, 3937499, 4148927, 4976639, 5242439, 5375999, 5467499, 5687999, 6219827, 6482699, 6560999, 7464959, 7654499, 7717499, 8294999, 8361359, 9525599, 9754919, 10238399, 10321919, 10450943, 10484879, 12287999, 12797999, 13590527, 13778099, 14117879, 14405999, 14751743, 14929919, 15308999, 15434999, 16589999, 16669799, 17860499, 18007499, 18370799, 18580799, 19509839, 22049999, 22396499, 26459999, 27433727, 29393279, 29647547, 29859839, 29861999, 31116959, 33179999, 33339599, 34401023, 34836479, 35720999, 36741599, 36863999, 39019679, 39321599, 39815999, 44099999, 44236799, 46807499, 47416319, 48009023, 49601159, 53343359, 55045619, 59723999, 59999999, 62128127, 62710199, 63787499, 63989999, 66679199, 68284439, 68802047, 69672959, 71441999, 74680703, 78643199, 82319999, 83348999, 85030559, 88942643, 90016919, 93297419, 96018047, 98018423. For p = 7 or prime p >= 59, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 7, 31 and 79, then 2^5 * 3 * 5^2 * 23^2 * 29 * p is a solution. E.g., p = 7, 59, 83, 107, 167, 179, 359, 383, 479, 503, 587, 719, 839, 863, 1439, 1487, 1619, 2099, 2879, 2999, 3023, 3779, 4703, 4799, 5399, 5879, 6047, 6719, 7559, 8747, 8819, 10079, 10799, 12347, 14699, 18959, 21599, 23039, 25919, 26459, 29399, 29759, 37799, 40499, 40823, 45503, 51839, 52919, 53759, 66359, 66959, 67499, 71999, 81647, 91139, 103787, 111599, 119447, 130199, 133919, 134999, 138239, 143999, 145823, 150659, 158759, 170099, 182279, 188159, 201599, 217727, 227519, 233279, 238079, 244943, 260399, 263423, 264599, 291647, 301319, 328103, 338687, 349919, 369023, 377999, 379199, 421847, 453599, 464999, 470399, 483839, 489887, 510299, 510383, 514499, 524243, 587999, 602639, 617147, 650327, 671999, 691199, 774143, 839999, 907199, 952559, 995327, 1008419, 1020599, 1028579, 1028999, 1049999, 1071359, 1180979, 1199999, 1210103, 1254203, 1270079, 1327199, 1474559, 1505279, 2041199, 2057999, 2142719, 2187359, 2303999, 2361959, 2551499, 2667167, 2678399, 2778299, 2787119, 2812319, 2915999, 2916479, 2949119, 3110399, 3145463, 3173903, 3185279, 3228959, 3281039, 3333119, 3344543, 3571199, 3719999, 3732479, 3736367, 3937499, 4148927, 4976639, 5142899, 5242439, 5375999, 5467499, 5687999, 6219827, 6482699, 6560999, 6562079, 6666239, 7464959, 7654499, 7717499, 8294999, 8361359, 9110279, 9113999, 9404159, 9525599, 9754919, 10238399, 10321919, 10450943, 10484879, 10499327, 10665983, 11427839, 12108599, 12287999, 12797999, 13284863, 13590527, 13778099, 14117879, 14405999, 14751743, 14929919, 15308999, 15434999, 16251563, 16589999, 16669799, 17084843, 17436383, 17860499, 17998847, 18007499, 18227999, 18370799, 18580799, 19373759, 19509839, 21092399, 22049999, 22396499, 23435999, 24115643, 26459999, 26569727, 27433727, 28569599, 29062499, 29393279, 29647547, 29859839, 29861999, 29890943, 31005827, 31116959, 32914559, 33179999, 33339599, 34177499, 34283519, 34401023, 34836479, 34872767, 35599283, 35720999, 36000299, 36455999, 36741599, 36863999, 38747519, 39019679, 39321599, 39815999, 44099999, 44236799, 45205439, 46807499, 47416319, 47457899, 48009023, 48758783, 49601159, 49996799, 50400419, 53343359, 54661679, 55045619, 58124999, 59723999, 59999999, 60263999, 60948479, 62128127, 62495999, 62710199, 63277199, 63787499, 63989999, 66679199, 68284439, 68567039, 68802047. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 11, 31 and 509, then 2^4 * 3 * 67187 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 263, 383, 863, 1187, 1487, 2903, 3167, 5807, 8747, 21383, 25343, 42767, 49103, 98207, 103787, 152063, 253703, 288683, 304127, 351383, 369023, 684287, 995327, 1486847, 2968487, 3736367, 6816527, 9427967, 12649823, 13284863, 15729647, 15844223, 25299647, 26569727, 29890943, 31049567, 35475263, 48758783, 53526527, 87503327, 92475107, 93148703, 130842623, 138141827, 280600847, 380633087, 404794367, 491170943, 567604223, 658243583, 752371223, 859123583, 935384207, 1012254407. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5 and 43, then 2 * 3 * 5 * 46439 * p is a solution. E.g., p = 7, 11, 23, 47, 59, 107, 179, 359, 383, 479, 719, 863, 1439, 1619, 2063, 2579, 2879, 2999, 4127, 4799, 5399, 8747, 10799, 12899, 21599, 23039, 25799, 25919, 40499, 51599, 51839, 66047, 67499, 71999, 110939, 116099, 134999, 138239, 143999, 233279, 309599, 349919, 691199, 990719, 995327, 1109399, 1180979, 1199999, 1474559, 1797227, 1857599, 2303999, 2361959, 2915999, 2949119, 3110399, 3732479, 4976639, 5467499, 5642459, 5990759, 6560999, 7464959, 9674999, 12287999, 14929919, 15851519, 18808199, 21300479, 28622519, 29859839, 36863999, 39321599, 42799103, 43537499, 44236799, 50155199, 53916839, 59999999, 78643199, 85030559, 96749999, 107833679, 179722799, 205128059, 214668899, 215667359, 223948799, 232199999, 255091679, 311039999, 340807679, 348299999, 359445599, 379687499, 491519999, 517601663, 566870399, 589823999, 622079999, 641986559, 759374999, 806249999, 863999999, 915920639, 1031999999, 1035203327, 1133740799, 1179647999, 1349999999, 1399679999, 1454112767, 1530550079, 1540767743, 2015624999, 2063999999, 2267481599. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 19 and 2341, then 2 * 3 * 533747 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 227, 383, 863, 1367, 1823, 8747, 17327, 701783, 886463, 995327, 3127703, 4269983, 10535423, 35487743, 68319743, 71803583, 79847423, 121694543, 382952447, 388343807, 909512063, 969348383. For prime p > 5 and != 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 31 and 8867, then 2^2 * 3 * 5^2 * 53201 * p is a solution. E.g., p = 7, 23, 47, 83, 107, 167, 383, 503, 587, 863, 1487, 3023, 4703, 6047, 8747, 12347, 40823, 81647, 103787, 145823, 217727, 244943, 263423, 291647, 328103, 338687, 369023, 421847, 489887, 510383, 638423, 774143, 995327, 1210103, 2667167, 3736367, 4148927, 10450943, 10499327, 10665983, 13284863, 14751743, 17084843, 17436383, 17998847, 24115643, 26569727, 27433727, 29647547, 29890943, 31005827, 34872767, 35599283, 48009023, 48758783, 74680703, 88942643, 96018047, 98018423, 119042783, 119593907, 125411327, 144657407. For prime p > 5 and != 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 31 and 953, then 2^4 * 3 * 51461 * p is a solution. E.g., p = 7, 23, 47, 107, 383, 863, 1487, 8747, 103787, 369023, 995327, 3736367, 13284863, 26569727, 29890943, 48758783, 118568447, 658243583, 1316487167. For prime p > 5 and != 47, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 37, 61 and 307, then 2 * 3 * 17^2 * 47^2 * p is a solution. E.g., p = 7, 11, 23, 107, 383, 863, 887, 8747, 8783, 14207, 35963, 89303, 265247, 408923, 511487, 591407, 632447, 995327, 2248703, 2913083, 4907087, 5447543, 5658623, 9047903, 9548927, 11317247, 24944363, 26433983, 36405287, 52867967, 59476463, 66518303, 111233987, 128910527, 158603903, 196111583, 465567743, 485720063, 577214207, 690352127, 931135487. For prime p > 5 and != 383, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 11 and 683, then 2^2 * 3 * 17 * 45077 * p is a solution. E.g., p = 7, 11, 23, 47, 83, 107, 167, 263, 503, 587, 863, 1187, 2903, 3023, 3167, 4703, 5807, 6047, 8747, 10163, 12347, 20327, 21383, 25343, 29567, 40823, 42767, 71147, 81647, 116423, 152063, 217727, 229487, 232847, 244943, 263423, 288683, 304127, 310463, 338687, 351383, 489887, 670823, 684287, 731807, 774143, 823283, 827903, 995327, 1210103, 1397087, 1486847, 1655807, 1788863, 2347883, 2667167, 2694383, 3577727, 4148927, 4967423, 5388767, 8964647, 10450943, 10843307, 12649823, 13172543, 13246463, 14751743, 14758127, 18213887, 20490623, 22030847, 22353407, 22819103, 25299647, 26492927, 27433727, 29647547, 29804543, 31049567, 48009023, 51342983, 53526527, 53552663, 62478107, 66012407, 71717183, 74680703, 87271007, 88942643, 92207807, 93148703, 96018047, 98018423, 111560063, 114960383, 119042783, 125411327, 130842623, 158070527, 169047647, 180787067, 201523247, 208631807, 222144383, 236130047, 268441403, 280600847, 301771007, 306561023, 359837183, 380633087, 386394623. For prime p > 5 and != 47, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, and 1381, then 2 * 3 * 447443 * p is a solution. E.g., p = 7, 11, 23, 107, 383, 863, 8747, 530303, 995327, 6363647, 205973387, 229091327. For p = 11 or prime p >= 83, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 11, 79 and 127, then 2^6 * 3 * 5 * 23^2 * 43 * p is a solution. E.g., p = 11, 83, 107, 167, 263, 383, 503, 587, 863, 1187, 1523, 2903, 3023, 3167, 4703, 5807, 6047, 8747, 10163, 10667, 12347, 20327, 21383, 25343, 29567, 32003, 40823, 42767, 45503, 64007, 71147, 81647, 116423, 119447, 152063, 166847, 217727, 232847, 244943, 263423, 288683, 291983, 304127, 310463, 338687, 351383, 352043, 489887, 524243, 603503, 650327, 670823, 684287, 704087, 731807, 774143, 823283, 827903, 864107, 995327, 1106423, 1210103, 1254203, 1357883, 1365503, 1397087, 1486847, 1655807, 1685543, 1788863, 2347883, 2667167, 2694383, 3145463, 3344543, 3577727, 4148927, 4443983, 4967423, 4978907, 5388767, 6219827, 6757343, 7153607, 8583167, 8964647, 10450943, 12649823, 13172543, 13246463, 13590527, 14751743, 14758127, 18213887, 20490623, 22312883, 22353407, 22482767, 22819103, 25299647, 26492927, 27433727, 29647547, 29804543, 31049567, 33035903, 34401023, 35396423, 43452287, 44625767, 48009023, 51342983, 51750467, 53526527, 53937407, 58271663, 62128127, 66012407, 68802047, 71717183, 74680703, 74747903, 79492643, 80500727, 88942643, 92207807, 93148703, 96018047, 98018423, 111560063, 114960383, 116543327, 119042783, 121100363. For p = 7 or prime p >= 23, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 31, 103 and 619, then 2^4 * 3 * 382541 * p is a solution. E.g., p = 7, 23, 47, 107, 383, 863, 1487, 8747, 9887, 29663, 79103, 103787, 344843, 369023, 995327, 1805003, 1901567, 2452223, 3736367, 13284863, 21414923, 24828767, 26569727, 28880063, 29890943, 48758783, 58948607, 64980143, 102518783, 157352687. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 23, and 31, then 2^4 * 3 * 137 * 3347 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 1487, 2207, 8747, 77003, 103787, 369023, 514187, 995327, 1180727, 2336063, 3656447, 3736367, 5432507, 9255383, 13284863, 26569727, 29890943, 38193983, 48758783, 74754047, 118278143. For prime p > 7, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, 29, and 43, then 2 * 3^2 * 5 * 29927 * p is a solution. E.g., p = 11, 23, 47, 107, 347, 383, 467, 863, 2027, 2063, 4127, 8423, 8747, 12527, 16223, 20123, 20183, 44543, 54287, 66047, 72383, 89087, 995327, 1314143, 1797227, 1819583, 1954367, 2267303, 3175847, 3471647, 4682687, 5252363, 7499543, 11625983, 12584723, 14999087, 18690047, 21835007, 23433623, 24564383, 36531647, 37672127, 42799103, 62489663, 106445663, 111181823, 165347327, 178906103, 197843567, 207028223, 250159103, 365249663, 422143487, 436620287, 461264543, 517601663, 541577087, 614526587, 1035203327, 1149523127, 1218548447. For p = 7 or prime p >= 23, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 31, and 16369, then 2^4 * 3 * 294641 * p is a solution. E.g., p = 7, 23, 47, 107, 383, 863, 1487, 8747, 103787, 369023, 995327, 3736367, 6089267, 13284863, 26569727, 29890943, 48758783, 127285343. For prime p > 5 and != 107 and != 479, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, and 7, then 2 * 3 * 5 * 1190699 * p is a solution. E.g., p = 7, 11, 23, 47, 59, 83, 167, 179, 359, 383, 503, 587, 719, 839, 863, 1439, 1619, 2099, 2879, 2999, 3023, 3779, 4703, 4799, 5399, 5879, 6047, 6719, 7559, 8747, 8819, 10079, 10799, 12347, 14699, 21599, 23039, 25919, 26459, 29399, 37799, 40499, 40823, 51839, 52919, 53759, 67499, 71999, 81647, 134999, 138239, 143999, 158759, 170099, 188159, 201599, 217727, 233279, 244943, 263423, 264599, 338687, 349919, 377999, 453599, 470399, 483839, 489887, 510299, 514499, 587999, 671999, 691199, 774143, 839999, 907199, 952559, 995327, 1008419, 1020599, 1028999, 1049999, 1180979, 1199999, 1210103, 1270079, 1474559, 1505279, 2041199, 2057999, 2303999, 2361959, 2551499, 2667167, 2778299, 2915999, 2949119, 3110399, 3732479, 3937499, 4148927, 4976639, 5375999, 5467499, 6482699, 6560999, 7464959, 7654499, 7717499, 9525599, 10321919, 10450943, 12287999, 13778099, 14117879, 14405999, 14751743, 14929919, 15308999, 15434999, 16669799, 17860499, 18007499, 18370799, 22049999, 26459999, 27433727, 29393279, 29647547, 29859839, 31116959, 33339599, 34836479, 35720999, 36741599, 36863999, 39321599, 44099999, 44236799, 47416319, 48009023, 49601159, 53343359, 59999999, 63787499, 66679199, 69672959, 71441999, 74680703, 78643199, 82319999, 83348999, 85030559, 88942643, 90016919, 96018047, 98018423. For prime p > 7, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, and 181, then 2^2 * 3 * 7 * 106427 * p is a solution. E.g., p = 11, 23, 47, 83, 107, 167, 383, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 40823, 81647, 217727, 243263, 244943, 263423, 338687, 410507, 417023, 486527, 489887, 774143, 995327, 1210103, 1459583, 1915703, 2667167, 3284063, 3538187, 4148927, 6568127, 10450943, 14751743, 27433727, 29647547, 30025727, 48009023, 74680703, 88669727, 88942643, 96018047, 98018423, 107279423, 119042783, 125411327, 214558847, 222905843. For prime p > 7, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, and 701, then 2 * 3 * 5 * 145807 * p is a solution. E.g., p = 11, 23, 47, 107, 383, 467, 863, 2027, 8423, 8747, 16223, 33647, 75707, 269183, 538367, 995327, 1314143, 1421627, 1819583, 18690047, 21835007, 24564383, 106445663, 136476287, 207028223, 363936767, 461264543, 697724927, 818857727. For prime p > 7, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, and 701, then 2 * 3^2 * 5 * 145807 * p is a solution. E.g., p = 11, 23, 47, 107, 383, 467, 863, 2027, 8423, 8747, 16223, 33647, 75707, 269183, 538367, 995327, 1314143, 1421627, 1819583, 18690047, 21835007, 24564383, 106445663, 136476287, 207028223. For prime p > 5 and != 23 and != 83 and != 383, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 127, and 293, then 2^6 * 3 * 172283 * p is a solution. E.g., p = 7, 11, 47, 107, 167, 503, 587, 863, 1523, 3023, 4703, 6047, 8747, 10667, 12347, 32003, 40823, 49223, 64007, 81647, 217727, 244943, 263423, 338687, 344567, 489887, 774143, 864107, 995327, 1210103, 1365503, 2667167, 3600383, 4148927, 4443983, 4725503, 10450943, 14751743, 27433727, 29647547, 48009023, 74680703, 75608063, 88942643, 96018047, 98018423. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 211, and 1429, then 2 * 3 * 1809113 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 8747, 22787, 617327, 995327, 9877247, 34192127, 50003567, 98017967. For p = 7 or prime p >= 23, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 31, and 37, then 2 * 3 * 5^2 * 174047 * p is a solution. E.g., p = 7, 23, 47, 83, 107, 167, 383, 503, 587, 863, 887, 1487, 3023, 4703, 6047, 8747, 12347, 14207, 27527, 35963, 40823, 65267, 81647, 103787, 145823, 217727, 244943, 263423, 291647, 328103, 338687, 369023, 371627, 421847, 489887, 510383, 511487, 591407, 774143, 995327, 1210103, 1527803, 2013983, 2667167, 2697743, 2913083, 3736367, 4074143, 4148927, 5946047, 9249407, 10450943, 10499327, 10665983, 10694627, 12332543, 13284863, 14751743, 15787307, 17084843, 17436383, 17998847, 18333647, 24115643, 26569727, 27433727, 29647547, 29890943, 31005827, 34872767, 35599283, 35676287, 37594367, 39443627, 42293663, 43163903, 48009023, 48758783, 53762183, 57957983, 74680703, 86327807, 88942643, 96018047, 98018423, 119042783, 119593907, 125411327. For prime p > 5 and != 23 and != 83 and != 383, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, and 293, then 2^2 * 3 * 17 * 172283 * p is a solution. E.g., p = 7, 11, 47, 107, 167, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 40823, 49223, 81647, 217727, 244943, 263423, 338687, 344567, 489887, 774143, 995327, 1210103, 2667167, 3600383, 4148927, 4725503, 10450943, 14751743, 27433727, 29647547, 48009023, 74680703, 75608063, 88942643, 96018047, 98018423. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, and 1009, then 2^3 * 3 * 181619 * p is a solution. E.g., p = 7, 11, 23, 47, 59, 107, 179, 359, 383, 479, 719, 863, 1439, 1619, 2879, 2999, 4799, 5399, 8747, 10799, 12107, 21599, 23039, 25919, 40499, 51839, 60539, 67499, 71999, 134999, 138239, 143999, 233279, 349919, 691199, 995327, 1162367, 1180979, 1199999, 1474559, 2303999, 2361959, 2915999, 2949119, 3110399, 3732479, 4976639, 5467499, 6560999, 7264799, 7464959, 12287999, 14711219, 14929919, 26153279, 29422439, 29859839, 36863999, 39321599, 44236799, 52306559, 59999999, 78643199, 85030559, 111587327, 223948799, 242159999, 255091679, 282455423, 311039999, 326915999, 371957759, 379687499, 484319999, 491519999, 566870399, 589823999, 622079999, 759374999. For p = 7 or prime p >= 23, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 31, and 1009, then 2 * 3 * 5^2 * 181619 * p is a solution. E.g., p = 7, 23, 47, 59, 107, 179, 359, 383, 479, 719, 863, 1439, 1487, 1619, 2879, 2999, 4799, 5399, 8747, 10799, 12107, 21599, 23039, 25919, 29759, 40499, 51839, 60539, 66959, 67499, 71999, 103787, 111599, 133919, 134999, 138239, 143999, 150659, 233279, 238079, 301319, 349919, 369023, 464999, 602639, 691199, 995327, 1071359, 1126043, 1162367, 1180979, 1199999, 1474559, 2142719, 2252087, 2303999, 2361959, 2678399, 2915999, 2949119, 3110399, 3571199, 3719999, 3732479, 3736367, 4976639, 5467499, 6560999, 6756263, 7264799, 7464959, 11427839, 12287999, 13284863, 14711219, 14929919, 26153279, 26569727, 28569599, 29062499, 29422439, 29859839, 29890943, 34283519, 36863999, 39321599, 44236799, 48758783, 52306559, 58124999, 59999999, 60263999, 60948479, 68567039, 74399999, 78643199, 83699999, 85030559, 97627679, 107135999, 110822519, 111587327, 120527999. For p = 7 or prime p >= 47, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, and 71, then 2 * 3 * 5 * 184031 * p is a solution. E.g., p = 7, 47, 107, 383, 863, 8747, 61343, 995327, 38654387, 609880343. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 61, and 97, then 2 * 3 * 2556143 * p is a solution. E.g., p = 7, 11, 23, 47, 107, 383, 863, 6983, 8747, 8783, 13967, 89303, 632447, 995327, 2248703, 4022783, 5447543, 8045567, 14452223, 25987463, 40898303, 196111583. For p = 7 or prime p >= 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, and 3049, then 2 * 3 * 5 * 237821 * p is a solution. E.g., p = 7, 107, 383, 467, 863, 2027, 8423, 8747, 16223, 995327, 1314143, 1819583, 3951503, 15220607, 18690047, 21835007, 24564383, 55650347, 106445663, 142254143, 207028223, 231162983, 461264543. For prime p > 5 and != 47, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, and 9181, then 2 * 3 * 2644127 * p is a solution. E.g., p = 7, 11, 23, 107, 383, 863, 8747, 991547, 995327, 95188607. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 29, and 151, then 2 * 3 * 5 * 262739 * p is a solution. E.g., p = 7, 11, 23, 47, 59, 107, 179, 347, 359, 383, 479, 719, 863, 1439, 1619, 2879, 2999, 3623, 4799, 5399, 7247, 8699, 8747, 10799, 12527, 18119, 20183, 21599, 23039, 25919, 26099, 40499, 41759, 44543, 50459, 51839, 62639, 67499, 71999, 89087, 134999, 138239, 143999, 151379, 181199, 208799, 233279, 302759, 349919, 420383, 489239, 521999, 587087, 652319, 691199, 840767, 978479, 995327, 1009199, 1113599, 1180979, 1199999, 1474559, 2303999, 2348999, 2361959, 2627399, 2899199, 2915999, 2949119, 3110399, 3732479, 3784499, 4682687, 4976639, 5467499, 6560999, 6959999, 7127039, 7464959, 7568999, 10022399, 11625983, 12287999, 14093999, 14929919, 15137999, 19376639, 25223039, 25229999, 28312499, 29859839, 35635199, 36531647, 36863999, 39321599, 43499999, 44236799, 44325143, 45100799, 48923999, 52199999, 59999999, 63422999, 78643199, 80713727, 85030559, 96215039, 97847999, 150294527, 158694959, 165347327, 174389759, 182917499, 195749999, 205725419, 223948799, 255091679, 264189599, 267263999, 311039999, 315201023, 371097599, 378345599, 379687499, 411450839, 417599999, 456645599. For prime p > 5, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 641, and 673, then 2^2 * 3 * 2588357 * p is a solution. E.g., p = 7, 11, 23, 47, 83, 107, 167, 383, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 15383, 40823, 81647, 113063, 217727, 244943, 263423, 338687, 489887, 774143, 995327, 1210103, 2667167, 3052727, 4070303, 4148927, 4361363, 5169023, 5435147, 6202367, 7876607, 10450943, 11630303, 14751743, 27433727, 29647547, 48009023, 74680703, 82704383, 88942643, 91588643, 96018047, 98018423, 119042783. For prime p > 5 and != 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, and 181, then 2^2 * 3 * 11 * 106427 * p is a solution. E.g., p = 7, 23, 47, 83, 107, 167, 383, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 40823, 81647, 217727, 243263, 244943, 263423, 338687, 410507, 417023, 486527, 489887, 774143, 995327, 1210103, 1459583, 1915703, 2667167, 3284063, 3538187, 4148927, 6568127, 10450943, 14751743, 27433727, 29647547, 30025727, 48009023, 74680703, 88669727, 88942643, 96018047, 98018423, 107279423, 119042783, 125411327, 214558847, 222905843. For prime p > 5 and != 11 and != 107 and != 383, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 7, 409, and 811, then 2 * 3 * 5 * 53^2 * 204371 * p is a solution. E.g., p = 7, 23, 47, 83, 167, 503, 587, 863, 3023, 4703, 6047, 8747, 12347, 40823, 81647, 87587. For p = 7 or prime p >= 107, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, and 3049, then 2 * 3 * 5^3 * 237821 * p is a solution. E.g., p = 7, 107, 383, 467, 863, 2027, 8423, 8747, 16223, 995327, 1314143, 1819583, 3951503, 15220607, 18690047. For prime p >= 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, 31, 53, and 701, then 2^4 * 3 * 5 * 953 * 145807 * p is a solution. E.g., p = 11, 23, 47, 107, 383, 467, 863, 1487, 1907, 2027, 5087, 8423, 8747, 16223, 33647, 75707. For prime p >= 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 13, 31, 53, and 701, then 2^4 * 3^2 * 5 * 953 * 145807 * p is a solution. E.g., p = 11, 23, 47, 107, 383, 467, 863, 1487, 1907, 2027, 5087, 8423, 8747, 16223. For prime p >= 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 5, 13, 127, and 701, then 2^6 * 3^2 * 5 * 499 * 145807 * p is a solution. E.g., p = 11, 23, 47, 59, 107, 179, 359, 383, 467, 479, 719, 863, 1439, 1523, 1619, 2027, 2879, 2999, 3119, 4679, 4799, 5399, 8423, 8747, 10799, 11699. For prime p >= 11, if (p-1)/2 is prime and p+1 has no prime factors other than 2, 3, 29, 43, and 1093, then 2 * 3^6 * 5 * 29927 * p is a solution. E.g., p = 11, 23, 47, 107, 347, 383, 863, 2063, 4127, 8747, 12527, 20183, 44543, 66047, 89087, 118043, 236087, 380363, 563987, 995327, 1797227, 3471647, 4682687, 11625983, 12584723.
Count = 21250
Near the end of sigma ( phi ( ) ) : From "5" to "5 figures"
is a description of 14000 solutions of the form 2^5 * 3 * 5^(p5-1) * 23^2 * 29 * p . The count 14000 is the product of 1400 and 10 , where 1400 is the number of suitable values of p with both sigmas <= 2^53 , and 10 was the number of proven primes of the form (5^p5-1) / 2^2 . Because ( 5^3407 - 1 ) / 4has been proven prime by Tom Wu ,
p5 can be any of 3 , 7 , 11 , 13 , 47 , 127 , 149 , 181 , 619 , 929 , or 3407 , and the number used in the computation should be not 10 , but rather 11 . Thus the number of solutions of the form 2^5 * 3 * 5^(p5-1) * 23^2 * 29 * p is at least 1400 * 11 = 15400 . ++++++++++++++++++++++++
The results infra are somewhat duplicative of those shown in Section 1.
For prime p >= 11 , if (p-1)/2 and (3^p3-1) / 2 are
prime , and p+1 has no prime factors other than 2 , 3 , 29 , and 43 ,
then 2 * 3^(p3-1) * 5 * 29927 * p is a solution .
There are 87 , exactly, suitable values of p with both sigmas <=
2^53, the least of which are p = 11 , 23 , 47 , 107 , 347 , 383 ,
863 , 2063 , 4127 , and 8747 . The greatest of these 87 least is
p = 8023504885383167 . Each of these 87 may be paired with a p3
taken from, at least, these 10 values: 3 , 7 , 13 , 71 , 103 , 541 ,
1091 , 1367 , 1627 , 4177 .
The least of the resulting 870 solutions has p3 = 3 and p = 11 and
is 2 * 3^2 * 5 * 29927 * 11 = 29627730 .
sigma ( phi (n) ) =
n phi ( n ) sigma ( n ) phi ( sigma (n) )
29627730 7182240 84037824 24385536
++++++++++++++++++++++++
For prime p >= 11 , if (p-1)/2 and (3^p3-1) / 2 are prime , and
p+1 has no prime factors other than 2 , 3 , 29 , 31 , 43 , and 53 ,
then 2^4 * 3^(p3-1) * 5 * 953 * 29927 * p is a solution .
There are 498 , exactly, suitable values of p with both sigmas <=
2^53, the least of which are p = 11 , 23 , 47 , 107 , 347 , 383 ,
863 , 1487 , 1907 , 2063 . The greatest of these 498 least is
p = 8023504885383167 . Each of these 498 may be paired with a p3
taken from, at least, these 10 values: 3 , 7 , 13 , 71 , 103 , 541 ,
1091 , 1367 , 1627 , 4177 .
The least of the resulting 4980 solutions has p3 = 3 and p = 11 and
is 2^4 * 3^2 * 5 * 953 * 29927 * 11 = 225881813520 .
sigma ( phi (n) ) =
n phi ( n ) sigma ( n ) phi ( sigma (n) )
225881813520 54699939840 828444868992 228248616960
2000 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11N25 , 11N80 , 11Y11 , 11Y55 , 11Y70
Walter Nissen
2008-10-15