%I #40 Aug 17 2023 08:16:58
%S 9,25,27,45,49,81,121,125,169,225,243,289,325,343,361,405,529,561,625,
%T 637,729,841,891,961,1105,1125,1225,1331,1369,1377,1681,1729,1849,
%U 2025,2187,2197,2209,2401,2465,2809,2821,3125,3321,3481
%N Weak Carmichael numbers.
%C 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
%H Amiram Eldar, <a href="/A225498/b225498.txt">Table of n, a(n) for n = 1..10000</a>
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1305.1867">Generalizations of Carmichael numbers I,</a> arXiv:1305.1867 [math.NT], May 4, 2013.
%H R. G. E. Pinch, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1202611-7">The Carminchael numbers up to 10^15</a>, Math. Comp. 61 (1993), 381-391.
%H R. G. E. Pinch, <a href="http://arxiv.org/abs/math/0604376">The Carmichael numbers up to 10^18</a>, arXiv:math/060437 [math.NT], 2006.
%p with(numtheory): isweakCarmichael := proc(n)
%p if irem(n, 2) = 0 or isprime(n) then return false fi;
%p factorset(n) subset factorset(Clausen(n-1, 1)) end: # A160014
%p select(isweakCarmichael, [$2..3500]); # _Peter Luschny_, May 21 2019
%t pf[n_] := FactorInteger[n][[All,1]];
%t Clausen[0, _] = 1; Clausen[n_, k_] := Times @@ (Select[Divisors[n],
%t PrimeQ[# + k] &] + k);
%t weakCarmQ[n_] := If[EvenQ[n] || PrimeQ[n], Return[False], pf[n] == (pf[n] ~Intersection~ pf[Clausen[n-1,1]])];
%t Select[Range[2,3500], weakCarmQ] (* _Jean-François Alcover_, Jun 03 2019 *)
%Y Cf. A002997, A160014.
%K nonn,easy
%O 1,1
%A _Jonathan Vos Post_, May 08 2013