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.

8911, 29341, 314821, 410041, 1024651, 1152271, 5481451, 10267951, 14913991, 15247621, 36765901, 64377991, 67902031, 133800661, 139952671, 178482151, 188516329, 299736181, 362569201, 368113411, 395044651, 532758241, 579606301, 612816751, 620169409, 625482001

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Nov 21 2019

Keywords

• nonn

Comments

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

All these numbers have an odd number of prime factors.

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).

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.

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002322, A002997, A056912.

Subsequence of A262043, A329538 is a subsequence.

Programs

Mathematica

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]