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A329799 Odd squarefree composite numbers k such that p-1 divides k-1 and p-1 does not divide (k-1)/2 for every prime p|k.

%I A329799 #10 Nov 21 2019 22:16:44
%S A329799 8911,29341,314821,410041,1024651,1152271,5481451,10267951,14913991,
%T A329799 15247621,36765901,64377991,67902031,133800661,139952671,178482151,
%U A329799 188516329,299736181,362569201,368113411,395044651,532758241,579606301,612816751,620169409,625482001
%N A329799 Odd squarefree composite numbers k such that p-1 divides k-1 and p-1 does not divide (k-1)/2 for every prime p|k.
%C A329799 Carmichael numbers k such that p-1 does not divide (k-1)/2 for every prime p|k.
%C A329799 All these numbers have an odd number of prime factors.
%C A329799 Conjecture: these are odd composite numbers k such that b^{(k-1)/2} == -1 (mod k) for some base b such that ord_{k}(b) = lambda(k).
%C A329799 Note that if q is an odd prime, then b^{(q-1)/2} == -1 (mod q) for all bases b such that ord_{q}(b) = lambda(q) = q-1.
%C A329799 It seems that there are no odd composite numbers m such that b^{(m-1)/2} == -1 (mod m) for all bases b such that ord_{m}(b) = lambda(m). Checked up to 2^64.
%H A329799 Amiram Eldar, <a href="/A329799/b329799.txt">Table of n, a(n) for n = 1..10000</a>
%t A329799 aQ[n_] := Module[{f = FactorInteger[n], p}, p = f[[;;,1]]; Length[p] > 1 && Max[f[[;;,2]]] == 1 && AllTrue[p, Divisible[n-1, #-1] && !Divisible[(n-1)/2, #-1] &]]; Select[Range[3, 2*10^7], aQ]
%Y A329799 Cf. A002322, A002997, A056912.
%Y A329799 Subsequence of A262043, A329538 is a subsequence.
%K A329799 nonn
%O A329799 1,1
%A A329799 _Amiram Eldar_ and _Thomas Ordowski_, Nov 21 2019