cp's OEIS Frontend

A081754 Numbers n such that the number of noncongruent solutions to x^(2^m) == 1 (mod n) is the same for any m>=1.

%I A081754 #19 Feb 16 2025 08:32:48
%S A081754 1,2,3,4,6,7,8,9,11,12,14,18,19,21,22,23,24,27,28,31,33,36,38,42,43,
%T A081754 44,46,47,49,54,56,57,59,62,63,66,67,69,71,72,76,77,79,81,83,84,86,88,
%U A081754 92,93,94,98,99,103,107,108,114,118,121,124,126,127,129,131,132,133,134
%N A081754 Numbers n such that the number of noncongruent solutions to x^(2^m) == 1 (mod n) is the same for any m>=1.
%C A081754 Numbers n such that the multiplicative group of residues modulo n does not contain C4 as a subgroup. Equivalently, numbers not divisible by 16 or by any primes of the form 4k+1. - _Ivan Neretin_, Aug 02 2016
%C A081754 From _Jianing Song_, Oct 18 2021: (Start)
%C A081754 Numbers k such that psi(k) = A002322(k) is not divisible by 4.
%C A081754 Numbers k such that there are an odd number of coprime squares modulo k, i.e., numbers k such that A046073(k) is odd. (End)
%H A081754 Ivan Neretin, <a href="/A081754/b081754.txt">Table of n, a(n) for n = 1..10000</a>
%H A081754 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModuloMultiplicationGroup.html">Modulo Multiplication Group</a>.
%p A081754 filter:= n -> n mod 16 <> 0 and not member(1,numtheory:-factorset(n) mod 4):
%p A081754 select(filter, [$1..1000]); # _Robert Israel_, Aug 02 2016
%t A081754 Select[Range@135, ! Divisible[#, 16] && FreeQ[Mod[FactorInteger[3 #][[All, 1]], 4], 1] &] (* _Ivan Neretin_, Aug 02 2016 *)
%o A081754 (PARI) isA081754(n) = if(n>2, znstar(n)[2][1]%4==2, 1) \\ _Jianing Song_, Oct 18 2021
%Y A081754 Cf. A060594, A002322, A046073.
%K A081754 nonn
%O A081754 1,2
%A A081754 _Benoit Cloitre_, Apr 08 2003