A192297 Lesser of pseudo twin primes to base 2.

561, 643, 645, 1103, 1905, 2465, 2699, 2819, 4369, 4371, 4679, 6599, 10259, 12799, 14489, 16703, 18719, 19949, 23001, 25759, 25761, 29339, 30119, 31607, 33151, 39863, 41039, 42797, 49139, 52631, 55243, 60701, 62743, 68099, 72883, 83663, 85487, 87249, 90749

Offset: 1

Author

Vladimir Shevelev, Oct 11 2011

Keywords

• nonn

Comments

We call numbers {k,k+2} pseudo twin primes to base 2 if at least one of them is composite, while 2^(k-1) == 1 (mod k) and 2^(k+1) == 1 mod (k+2).

4369 is the only known term such that both k and k+2 are composite (cf. A173619). - Jianing Song, Nov 20 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..15587 (terms below 10^12; terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A001567, A002997, A141232.

Cf. A173619.

Programs

Maple

a:= proc(n) option remember; local k;
      for k from 2+`if` (n=1, 1, a(n-1)) by 2 while
        isprime(k) and isprime(k+2) or
          (2&^(k-1) mod k)<>1 or (2&^(k+1) mod (k+2))<>1
      do od; k
    end:
seq (a(n), n=1..40);  # Alois P. Heinz, Oct 13 2011

Mathematica

fQ[n_] := (! PrimeQ[n] || ! PrimeQ[n + 2]) && PowerMod[2, n - 1, n] == 1 && PowerMod[2, n + 1, n + 2] == 1; Select[2 Range@ 32000 + 1, fQ] (* Robert G. Wilson v, Oct 11 2011 *)

PARI

is(n)=Mod(2,n^2+2*n)^(n+2)==3*n+8 && (!isprime(n) || !isprime(n+2)) && n>1 \\ Charles R Greathouse IV, Dec 02 2014

Formula

2^(a(n) + 2) == 3*a(n) + 8 (mod a(n)*(a(n)+2)).

4*(2^(a(n)-1)-1) == -a(n)*((a(n)-1)/2) (mod a(n)*(a(n)+2)). - Davide Rotondo, Nov 07 2021