A173619 -id:A173619 - cp's OEIS frontend

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

Vladimir Shevelev, Oct 11 2011

nonn

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

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

Cf. A001567, A002997, A141232.

Cf. A173619.

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

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 *)

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

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

A335326 Lesser of 2 consecutive Fermat pseudoprimes to base 2 with no prime numbers in between them.

Original entry on oeis.org

4369, 13741, 31609, 6973057, 208969201

Offset: 1

Amiram Eldar, Jun 01 2020

The corresponding larger counterparts are 4371, 13747, 31621, 6973063, 208969223.
The corresponding gaps between the consecutive pseudoprimes are 2, 6, 12, 6, 22.
a(6) > 10^15, if it exists.

			4369 is a term since it is a Fermat pseudoprime to base 2, and between it and the next Fermat pseudoprime to base 2, 4371, there is only one number, 4370, which is composite.

Cf. A001567, A173619, A175736, A175737, A175738.

psp = Cases[Import["https://oeis.org/A001567/b001567.txt", "Table"], {, }][[;; , 2]]; s = {}; Do[If[NextPrime[psp[[k]]] > psp[[k + 1]], AppendTo[s, psp[[k]]]], {k, 1, Length[psp] - 1}]; s

cp's OEIS Frontend

A192297 Lesser of pseudo twin primes to base 2.

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