[seqfan] Re: Q about A152926

[Richard Mathar]

zs> From: zak seidov <zakseidov at yahoo.com>
zs> To: seqfaneu <seqfan at seqfan.eu>
zs> Subject: [seqfan]  Q about A152926
zs> 
zs> %C A152926 All terms == 6 (mod 15)
zs> - but why?
zs> thx, zak
zs> 
zs> %I A152926
zs> %S A152926 171,3801,5781,8721,8781,17601,18231,19011,24741,28251,40431,48951,
zs> %T A152926 49371,58821,70521,79401,79701,83391,87321,95781,96501,99501,102861,
zs> %U A152926 109431,123171,125061,137091,177201,220311,224511,225561,229551,242451
zs> %N A152926 Numbers n with property that 19n+{2,4, 8,10} are two subsequent twin primes.
zs> %C A152926 All terms == 6 (mod 15).
zs> %e A152926 19*171+{2,4}={3251,3253} and 19*171+{8,10}={3257,3259} are 85th and 86th twin primes.
zs> %e A152926 19*3801+{2,4}={72221,72223} and 19*3801+{8,10}={72227,72229} are 935-th and 936-th twin primes.
zs> %Y A152926 A001359 Lesser of twin primes.

All combinations but n = 6 ( mod 15) lead to conflicts with the requirements:

If n=15k, then 19n+10 = 285k+10 is divisible by 5 (not a prime).
If n=15k+1, then 19n+2= 285k+21 is divisible by 3 (not prime).
If n=15k+2, then 19n+2= 285k+40 is divisible by 5 (not prime).
If n=15k+3, then 19n+8= 285k+65 is divisible by 5 (not prime).
If n=15k+4, then 19n+2= 285k+78 is divisible by 3 (not prime).
If n=15k+5, then 19n+4= 285k+99 is divisible by 3 (not prime).
If n=15k+6, all requirements are met
If n=15k+7, then 19n+2= 285k+135 is divisible by 15 (not prime).
If n=15k+8, then 19n+4= 285k+156 is divisible by 3 (not prime).
If n=15k+9, then 19n+4= 285k+175 is divisible by 5 (not prime).
If n=15k+10, then 19n+2= 285k+192 is divisible by 3 (not prime).
If n=15k+11, then 19n+4= 285k+213 is divisible by 3 (not prime).
If n=15k+12, then 19n+2= 285k+230 is divisible by 5 (not prime).
If n=15k+13, then 19n+2= 285k+249 is divisible by 3 (not prime).
If n=15k+14, then 19n+4= 285k+270 is divisible by 15 (not prime).

So the only variant to get 4 primes 19n+2, 19n+4, 19n+8 and 19n+10
is n=15k+6 in these modulo classes.

Richard Mathar