A225498 Weak Carmichael numbers.

9, 25, 27, 45, 49, 81, 121, 125, 169, 225, 243, 289, 325, 343, 361, 405, 529, 561, 625, 637, 729, 841, 891, 961, 1105, 1125, 1225, 1331, 1369, 1377, 1681, 1729, 1849, 2025, 2187, 2197, 2209, 2401, 2465, 2809, 2821, 3125, 3321, 3481

Offset: 1

Author

Jonathan Vos Post, May 08 2013

Keywords

• nonn
• easy

Comments

An odd composite number n > 1 is a weak Carmichael number iff the prime factors of n are a subset of the prime factors of Clausen(n-1,1) (cf. A160014). If additionally n divides Clausen(n-1,1) then n is a Carmichael number. - Peter Luschny, May 21 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], May 4, 2013.
• R. G. E. Pinch, The Carminchael numbers up to 10^15, Math. Comp. 61 (1993), 381-391.
• R. G. E. Pinch, The Carmichael numbers up to 10^18, arXiv:math/060437 [math.NT], 2006.

Crossrefs

Cf. A002997, A160014.

Programs

Maple

with(numtheory): isweakCarmichael := proc(n)
if irem(n, 2) = 0 or isprime(n) then return false fi;
factorset(n) subset factorset(Clausen(n-1, 1)) end: # A160014
select(isweakCarmichael, [$2..3500]); # Peter Luschny, May 21 2019

Mathematica

pf[n_] := FactorInteger[n][[All,1]];
Clausen[0, ] = 1; Clausen[n, k_] := Times @@ (Select[Divisors[n],
PrimeQ[# + k] &] + k);
weakCarmQ[n_] := If[EvenQ[n] || PrimeQ[n], Return[False], pf[n] == (pf[n] ~Intersection~ pf[Clausen[n-1,1]])];
Select[Range[2,3500], weakCarmQ] (* Jean-François Alcover, Jun 03 2019 *)