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A345331 Odd numbers k > 1 such that m^(2^v(k-1)+1) == -m (mod k) has more than one solution modulo k, where v(k) = A007814(k) is the 2-adic valuation of k.

%I A345331 #17 Jun 14 2021 14:21:18
%S A345331 15,35,39,51,55,75,85,87,91,95,111,115,119,123,135,143,153,155,159,
%T A345331 175,183,187,195,203,205,215,219,221,235,247,255,259,267,275,287,291,
%U A345331 295,299,303,315,319,323,327,335,339,351,355,357,365,371,375,391,395,403
%N A345331 Odd numbers k > 1 such that m^(2^v(k-1)+1) == -m (mod k) has more than one solution modulo k, where v(k) = A007814(k) is the 2-adic valuation of k.
%C A345331 Note that for even k, m == -1 (mod k) is a solution.
%C A345331 All terms are composite.
%C A345331 Odd composite k is a term if and only if v(p-1) > v(k-1) for some prime factors p of k. See A345330 for a proof.
%C A345331 This sequence and the Carmichael numbers (A002997) are disjoint: if k is a Carmichael number, then p-1 | k-1 for all prime factors p.
%H A345331 Jianing Song, <a href="/A345331/b345331.txt">Table of n, a(n) for n = 1..10000</a>
%e A345331 51 is a term since 51 = 3 * 17 and v(17-1) = 4 > v(51-1) = 1. Also, m^(2^v(51-1)+1) == -m (mod 51) has three solutions: m == 0, 21, 30 (mod 51).
%o A345331 (PARI) isA345331(n) = if(!isprime(n) && n>1 && n%2, my(f=factor(n), w=omega(n)); for(i=1, w, if(valuation(f[i, 1]-1, 2) > valuation(n-1, 2), return(1))); 0, 0)
%Y A345331 Cf. A007814, A002997.
%Y A345331 Complement of A345330 with respect to A071904.
%K A345331 nonn
%O A345331 1,1
%A A345331 _Jianing Song_, Jun 14 2021