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Odd numbers n such that n is not coprime to the multiplicative order of 2 mod n.
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%I #34 Oct 15 2022 05:00:55

%S 9,21,25,27,39,45,49,55,57,63,75,81,99,105,111,117,121,125,135,147,

%T 153,155,165,169,171,175,183,189,195,201,203,205,207,219,225,231,237,

%U 243,245,253,261,273,275,279,285,289,291,297,301,305,309,315,325,327

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

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

%C 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.

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

%C All odd multiples of members of A273202. - _Robert Israel_, Jul 28 2016

%C 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

%H Robert Israel, <a href="/A274720/b274720.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%t A274720Q = OddQ[#] && ! CoprimeQ[MultiplicativeOrder[2, #], #] &; Select[Range[200], A274720Q] (* _JungHwan Min_, Jul 29 2016 *)

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

%Y Subset of A139392.

%Y Cf. A001220, A002326, A273202.

%K nonn

%O 1,1

%A _Robert Israel_, Jul 27 2016