OFFSET
1,1
COMMENTS
If p divides k, both (p-1)/2 and (p+1)/2 divide k-1.
Pinch (2007) called these numbers "Lucas-Carmichael-(-) numbers".
A term is called "strong Lucas-Carmichael-(-) number" if both p-1 and p+1 divide k-1. a(n) is strong for n = 2, 3, 6, 7, 8, 9, 12, 13, 15, 17, 19, 20, ... .
A term is called "unusually strong Lucas-Carmichael-(-) number" if p^2-1 divides k-1. a(n) is unusually strong for 2, 6, 7, 8, 9, 19, ... .
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..20
Mauro Fiorentini, Lucas-Carmichael (-) (numeri di) (in Italian).
Richard G. E. Pinch, Absolute quadratic pseudoprimes, Proceedings Conference on Algorithmic Number Theory, Turku, May 2007, TUCS General Publications, Vol. 46 (2007), pp. 113-128; alternative link.
MATHEMATICA
isok[k_] := k > 1 && OddQ[k] && !PrimeQ[k] && Module[{f = FactorInteger[k], p}, Max[f[[;; , 2]]] == 1 && AllTrue[f[[;; , 1]], Divisible[k-1, #-1] && Divisible[k-1, (#+1)/2] &]];
PROG
(PARI) isok(k) = if(k == 1 || isprime(k) || !issquarefree(k), 0, my(p = factor(k)[, 1]); for(i = 1, #p, if((k-1) % (p[i]-1) || (k-1) % ((p[i]+1)/2), return(0))); 1);
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 16 2026
STATUS
approved