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

1,1

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.

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 *)

Crossrefs

Cf. A002997, A160014.

Sequence in context: A155109 A268576 A053850 * A020210 A275196 A325373

Adjacent sequences: A225495 A225496 A225497 * A225499 A225500 A225501

Keyword

nonn,easy

Author

Jonathan Vos Post, May 08 2013

Status

approved