A194231 Numbers k such that at least one of k and k+2 is composite, while for every b coprime to k*(k+2), b^(k-1) == 1 (mod k) and b^(k+1) == 1 (mod k+2).

561, 1103, 2465, 2819, 6599, 29339, 41039, 52631, 62743, 172079, 188459, 278543, 340559, 488879, 656599, 656601, 670031, 1033667, 1909001, 2100899, 3146219, 5048999, 6049679, 8719307, 10024559, 10402559, 10877579, 11119103, 12261059, 14913989, 15247619

Offset: 1

Author

Vladimir Shevelev, Oct 12 2011

Keywords

• nonn

Comments

These might be called "Carmichael pseudo-twin-primes".

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002997, A001567, A141232.

Subsequences: A272754, A290692.

Programs

Maple

with(numtheory):
ic:= proc(n) local p;
       if not issqrfree(n) then false
     else for p in factorset(n) do
            if irem (n-1, p-1)<>0 then return false fi
          od; true
       fi
     end:
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 not (ic(k) and ic(k+2))
      do od; k
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Oct 12 2011

Mathematica

terms = 31; bMax = 20(* sufficient for 31 terms *); coprimes[n_] := Select[ Range[bMax], CoprimeQ[#, n]&]; Reap[For[n = m = 1, m <= terms, n += 2, If[CompositeQ[n] || CompositeQ[n+2], If[AllTrue[coprimes[n(n+2)], PowerMod[#, n-1, n] == 1 && PowerMod[#, n+1, n+2] == 1&], Print["a(", m, ") = ", n]; Sow[n]; m++]]]][[2, 1]] (* Jean-François Alcover, Mar 28 2017 *)

Formula

For every b coprime to a(n)*(a(n)+2), 2*b^(a(n)+1) == (b^2-1)*(a(n)+2) (mod a(n)*(a(n)+2)). Conversely (Max Alekseyev), if for every b coprime to N*(N+2), 2*b^(N+1) == (b^2-1)*(N+2) (mod N*(N+2)), then N is in the sequence. - Vladimir Shevelev, Oct 14 2011