login
Numbers n such that n is not the power of a prime and such that for every prime divisor p of n, p-1 divides n-1.
8

%I #11 Sep 22 2019 07:47:02

%S 45,225,325,405,561,637,891,1105,1125,1225,1377,1729,2025,2465,2821,

%T 3321,3645,3751,3825,4225,4961,5589,5625,6517,6525,6601,7381,8125,

%U 8281,8625,8911,9801,10125,10585,10625,12025,13357,13833,14161,15841,15925

%N Numbers n such that n is not the power of a prime and such that for every prime divisor p of n, p-1 divides n-1.

%C Weak Carmichael numbers that are not prime powers [Mestrovic] - _N. J. A. Sloane_, Aug 25 2013

%C The values of this sequence together with the set of prime powers (A000961) forms the sequence A087441.

%H Amiram Eldar, <a href="/A087442/b087442.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:1305.1867v1 [math.NT], May 4, 2013.

%e 45 is in A087442 because it is not a prime power and because its prime divisors are 3 and 5 and 3-1=2 as well as 5-1=4 divide 45-1 = 44.

%t wcQ[n_] := Length[(p = FactorInteger[n][[;;,1]])] > 1 && AllTrue[p, Divisible[n-1, #-1] &]; Select[Range[10^4], wcQ] (* _Amiram Eldar_, Sep 22 2019 *)

%o (GAP) for i in [2..20000] do if IsSubset (DivisorsInt(i-1), Set (FactorsInt (i)) - 1) and not IsPrimePowerInt (i) then Print (i, ", "); fi; od;

%Y Cf. A000961, A087441, A225498.

%K easy,nonn

%O 1,1

%A _Jens Voß_, Sep 04 2003