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A274720
Odd numbers n such that n is not coprime to the multiplicative order of 2 mod n.
2
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
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
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.
Sequence in context: A138786 A305733 A139392 * A338318 A322336 A327685
KEYWORD
nonn
AUTHOR
Robert Israel, Jul 27 2016
STATUS
approved