A367230 Base-2 Fermat pseudoprimes k such that the multiplicative order of 2 modulo k is odd.

2047, 4681, 15841, 42799, 52633, 90751, 220729, 256999, 271951, 486737, 514447, 647089, 741751, 916327, 1082401, 1145257, 1730977, 1969417, 2205967, 2304167, 2748023, 2811271, 2953711, 2976487, 3567481, 4188889, 4469471, 4835209, 4863127, 5016191, 5049001, 5681809

Offset: 1

Author

Amiram Eldar, Nov 11 2023

Keywords

• nonn

Comments

The corresponding sequence for primes is A014663.

These pseudoprimes seem to be relatively rare: among the 118968378 base-2 Fermat pseudoprimes below 2^64 only 6292535 are terms of this sequence.

These pseudoprimes appear in a theorem by Rotkiewicz and Makowski (1966) about pseudoprimes that are products of two Mersenne numbers (see A367229).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andrzej Rotkiewicz and Andrzej Makowski, On Pseudoprime Numbers of the Form M_p M_t, Elemente der Mathematik, Vol. 21 (1966), pp. 133-134.

Crossrefs

Intersection of A001567 and A036259.

A367231 is a subsequence.

Cf. A002326, A014663, A306413, A367229.

Programs

Mathematica

Select[2*Range[10^6] + 1, PowerMod[2, # - 1, #] == 1 && CompositeQ[#] && OddQ[MultiplicativeOrder[2, #]] &]

PARI

is(n) = n > 1 && n % 2 && Mod(2, n)^(n-1) == 1 && !isprime(n) && znorder(Mod(2, n)) % 2;