cp's OEIS Frontend

A333314 Composite non-Carmichael numbers k such that rad(phi(k)) divides k-1, where rad(k) is the squarefree kernel of k (A007947) and phi is the Euler totient function (A000010).

%I A333314 #10 Mar 15 2020 05:07:08
%S A333314 15,51,85,91,133,247,255,259,435,451,481,511,595,679,703,763,771,949,
%T A333314 1111,1141,1261,1285,1351,1387,1417,1615,1695,1843,1891,2047,2071,
%U A333314 2091,2119,2431,2509,2701,2761,2955,3031,3097,3145,3277,3367,3409,3589,3655,3667
%N A333314 Composite non-Carmichael numbers k such that rad(phi(k)) divides k-1, where rad(k) is the squarefree kernel of k (A007947) and phi is the Euler totient function (A000010).
%C A333314 McNew and Wright proved that this sequence is infinite.
%H A333314 Amiram Eldar, <a href="/A333314/b333314.txt">Table of n, a(n) for n = 1..10000</a>
%H A333314 Nathan McNew and Thomas Wright, <a href="https://doi.org/10.1142/S1793042116501153">Infinitude of k-Lehmer numbers which are not Carmichael</a>, International Journal of Number Theory, Vol. 12, No. 7 (2016), pp. 1863-1869; <a href="https://arxiv.org/abs/1508.05547">preprint</a>, arXiv:1508.05547 [math.NT], 2015.
%e A333314 15 = 3 * 5 is a term since it is composite and not a Carmichael number, and rad(phi(15)) = rad(8) = 2 divides 15 - 1 = 14.
%t A333314 rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); Select[Range[4000], Divisible[#-1, rad[EulerPhi[#]]] && !Divisible[#-1, CarmichaelLambda[#]] &]
%Y A333314 Complement of the primes and Carmichael numbers (union of A000010 and A002997) with respect to A187731.
%Y A333314 Complement of A002997 with respect to A238574.
%Y A333314 Cf. A007947, A238575.
%K A333314 nonn
%O A333314 1,1
%A A333314 _Amiram Eldar_, Mar 14 2020