A329799 - OEIS

%I #10 Nov 21 2019 22:16:44

%S 8911,29341,314821,410041,1024651,1152271,5481451,10267951,14913991,

%T 15247621,36765901,64377991,67902031,133800661,139952671,178482151,

%U 188516329,299736181,362569201,368113411,395044651,532758241,579606301,612816751,620169409,625482001

%N 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 Carmichael numbers k such that p-1 does not divide (k-1)/2 for every prime p|k.

%C All these numbers have an odd number of prime factors.

%C 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 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 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 Amiram Eldar, <a href="/A329799/b329799.txt">Table of n, a(n) for n = 1..10000</a>

%t 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 Cf. A002322, A002997, A056912.

%Y Subsequence of A262043, A329538 is a subsequence.

%K nonn

%O 1,1

%A _Amiram Eldar_ and _Thomas Ordowski_, Nov 21 2019