OFFSET
1,1
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
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar and Thomas Ordowski, Nov 21 2019
STATUS
approved