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

1,1

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)

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

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 *)

Prog

(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

Crossrefs

Cf. A001567, A002997, A141232.

Cf. A173619.

Sequence in context: A344673 A344706 A259172 * A080747 A306487 A074380

Adjacent sequences: A192294 A192295 A192296 * A192298 A192299 A192300

Keyword

nonn

Author

Vladimir Shevelev, Oct 11 2011

Status

approved