Augmentation of table of abundancies and non-abundancies which appeared
in :
Abundancy "Outlaws" of the Form sigma(N)+t / N
William G. Stanton and Judy A. Holdener
Journal of Integer Sequences, Vol. 10(2007), Article 07.9.6 ,
pp. 1-19 .
Within the list of blue , green , red , and black abundancies and
non-abundancies , pp. 13-18 ,
the abundancies shown in the table infra are coded as
black = abundancy status unknown , but ultimately are seen to be
blue = known to be an abundancy , by extending the reach of the search
from 10^6 to 2^34 .
The computation was structured to make this table exhaustive over that
range .
No member of this table is of the form
sigma ( p ) + 1 p + 2
----------------- = ----- .
p p
This is entirely consistent with the report of a somewhat similar
computation to 10^16 with a similar result in D (Q.1) in
With the identification here of 9 / 2 as an abundancy , the only
remaining black = unknown entries among the first 29 entries of the
list , through 10 / 3 , are of the form
p+2 / p , namely 5 / 3 , 7 / 5 , and 9 / 7 ,
corresponding to the first 3 odd primes .
Table .
abundancy ( N ) N 9 / 2 8910720 13 / 3 18506880 14 / 3 208565280 19 / 4 746444160 19 / 8 4840192 23 / 6 1907020800 24 / 5 668304000 25 / 6 2836487808 30 / 7 2056320 31 / 7 5322240 32 / 7 164989440 33 / 7 67858560 33 / 8 9694080 33 / 13 1444352 34 / 7 5228496000 34 / 9 2678400 35 / 8 25159680 35 / 9 7448760 35 / 16 13137664 37 / 8 10178138880 37 / 9 17660160 37 / 10 1962240 37 / 11 4316928 38 / 9 2744280 40 / 9 52141320 43 / 10 4260372480 43 / 11 1544970240 43 / 12 140451840 43 / 13 284024832 43 / 14 21848064 43 / 16 3121152 44 / 13 4333056 45 / 11 3231360 49 / 11 378069120 49 / 12 29795040 50 / 13 5346432 51 / 11 5765760 51 / 19 1884800 52 / 11 203575680 54 / 11 98017920 54 / 13 2970240 55 / 13 176432256 55 / 14 13571712 55 / 16 1938816 56 / 11 2646483840 57 / 13 2751840 57 / 20 24200960 57 / 22 13443584 57 / 23 111324416 60 / 13 26732160 61 / 13 3127662720 61 / 15 8555880960 61 / 18 1711176192 62 / 13 69189120 63 / 16 1272960 64 / 13 2144862720 64 / 15 2031120 64 / 27 6517665 65 / 14 2461415040 65 / 16 1935360 65 / 17 123070752 65 / 18 3701376 66 / 13 882161280 66 / 19 4523520 66 / 23 1095168 66 / 31 1386831872 68 / 15 39312000 68 / 19 5654400 69 / 16 373201920 69 / 19 4025932800 69 / 20 211891200 70 / 17 126628920 70 / 31 407267584 73 / 15 10456992000 73 / 16 139426560 73 / 17 4112640 73 / 25 1388800 74 / 17 300222720 74 / 19 37282560 74 / 23 9026304 74 / 25 101382400 74 / 31 4055296 75 / 16 3346237440 75 / 19 5581440 75 / 23 1351296 76 / 17 46652760 76 / 25 3751148800 76 / 31 150045952 77 / 16 830269440 77 / 18 14778339840 77 / 19 2708640 78 / 17 615353760 80 / 17 886402440 80 / 41 4466007 81 / 17 296281440 81 / 25 24752000 83 / 25 381709200 83 / 31 15268368 84 / 17 3545609760 84 / 19 2489760 85 / 18 834261120 85 / 19 9959040 86 / 21 983162880 86 / 23 3230392320 86 / 25 2418892800 86 / 31 96755712 88 / 19 18960480 88 / 25 9585000 88 / 31 4160495616 89 / 19 1832463360 89 / 20 96445440 89 / 21 22256640 89 / 22 34974720 89 / 23 4493368320 89 / 24 3179520 89 / 25 54758400 89 / 26 6429696 89 / 31 2190336 90 / 19 169303680 90 / 23 6756480 91 / 19 11950848000 91 / 20 628992000 91 / 22 29082240 91 / 23 60808320 91 / 24 2643840 92 / 21 13349145600 93 / 19 438197760 93 / 20 23063040 93 / 22 16156800 93 / 23 17487360 93 / 26 1537536 95 / 21 364989240 95 / 22 3705077376 95 / 23 3433643136 95 / 24 149288832 96 / 19 12697776000 97 / 61 5659641 98 / 23 685285920 98 / 25 1404000 98 / 61 548985177 99 / 23 222963840 99 / 25 2607120000 99 / 29 6904320 99 / 37 55790080 99 / 38 1507840 99 / 41 4555264 100 / 21 10219698720 100 / 27 10428264 100 / 31 1821312count = 149
Walter Nissen
2000 Mathematics Subject Classification : Primary 11A25 ; Secondary 11Y55 , 11Y70
initially written 2008-01-26
comments updated 2008-03-10