A274720 - OEIS

A274720

Odd numbers n such that n is not coprime to the multiplicative order of 2 mod n.

9, 21, 25, 27, 39, 45, 49, 55, 57, 63, 75, 81, 99, 105, 111, 117, 121, 125, 135, 147, 153, 155, 165, 169, 171, 175, 183, 189, 195, 201, 203, 205, 207, 219, 225, 231, 237, 243, 245, 253, 261, 273, 275, 279, 285, 289, 291, 297, 301, 305, 309, 315, 325, 327

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OFFSET

1,1

COMMENTS

Odd numbers n such that gcd(n, A002326((n-1)/2)) > 1.

A prime power p^k is in the sequence unless 2^(p-1) == 1 (mod p^k). In particular, for p^2 to not be in the sequence p must be a Wieferich prime.

If n is in the sequence, then so is every odd multiple of n.

All odd multiples of members of A273202. - Robert Israel, Jul 28 2016

Let G(i, j) = gcd(2^j - 1, j^(2^i) - 1). I conjectured that an odd positive integer n is a term of this sequence if and only if n is not of the form G(i, j). Jinyuan Wang (pers. comm.) proved the direct implication and the fact that, if n is not a term of this sequence, then n divides G(i, j) for some i and j. - Lorenzo Sauras Altuzarra, Sep 04 2022

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

25 is in the sequence because the order of 2 mod 25 is 20 and gcd(20,25)=5>1.

MAPLE

remove(t -> igcd(t, numtheory:-order(2, t))=1, [seq(i, i=3..1000, 2)]);

MATHEMATICA

A274720Q = OddQ[#] && ! CoprimeQ[MultiplicativeOrder[2, #], #] &; Select[Range[200], A274720Q] (* JungHwan Min, Jul 29 2016 *)

PROG

(PARI) is(n) = n%2!=0 && gcd(n, znorder(Mod(2, n))) > 1 \\ Felix Fröhlich, Jul 27 2016

CROSSREFS

Subset of A139392.