A formula generates some primitive friendly pairs = pfps .
This is a sequel to
Primitive Friendly Integers and Exclusive Multiples
where some terms are defined .
If all of the following are prime :
p
prime_1 = 2 - 1
p+2
prime_2 = 2 - 1
p+2
2 + 1
prime_3 = -------
3
then p and p+2 are a pair of twin primes and
2p-1 2p+3
2 * prime_1 and 2 * prime_2 * prime_3 form a pfp
p
2 + 1 1
with abundancy = ------ = 2 + ----
p-1 p-1
2 2
In the formula, cases p = 3 , 5 , and 17 yield :
pf1 pf1 pf2 pf2 abund ( pf )
224 5001 174592 90001000001 3*3 / 4
15872 90000000001 44736512 D ;43;127 3*11 / 16
1125891316908032 X;131071 12592977287606574252032 b;174763;524287 3*43691/65536
prime_1 and prime_2 are Mersenne primes generated by a pair of twin
primes .
How does the frequency of occurrence of these pfps compare to that of
Fermat primes ?
Footnote :
{ X;131071 ; b;174763;524287 } multiplied by
{ 0A001;23;107;3851 ; 04003;31;61 }
produces a pfp found by comparing 2 4-friendlies ,
i.e. , 2 multiperfects with abundancy = 4 ,
and removing common factors which occur with like multiplicities .
2^33 3^10 7 11 23 83 107 331 3851 43691 131071
2^37 3^4 7 11^3 31 61 83 331 43691 174763 524287
XA0110001;83;107;331;3851;43691;131071
b4013;31;61;83;331;43691;174763;524287
XA0010001;107;3851;131071 b4003;31;61;174763;524287
2000 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11N25
Walter Nissen
2008-07-13