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 A342256 Numbers k such that gcd(k, Phi_k(a)) > 1 for some a, where Phi_k is the k-th cyclotomic polynomial. 3

%I #10 Mar 08 2021 15:28:32

%S 2,3,4,5,6,7,8,9,10,11,13,14,16,17,18,19,20,21,22,23,25,26,27,29,31,

%T 32,34,37,38,39,41,42,43,46,47,49,50,52,53,54,55,57,58,59,61,62,64,67,

%U 68,71,73,74,78,79,81,82,83,86,89,93,94,97,98,100,101

%N Numbers k such that gcd(k, Phi_k(a)) > 1 for some a, where Phi_k is the k-th cyclotomic polynomial.

%C Indices of columns of A342255 with some elements greater than 1.

%C For k > 1, let p be the largest prime factor of k, then k is a term if and only if k = p^e*d with d | (p-1). See A342255 for more information.

%C Also numbers k such that A342257(k) > 1.

%H Jianing Song, <a href="/A342256/b342256.txt">Table of n, a(n) for n = 1..10000</a>

%F Equals Union_{p prime} (Union_{d|(p-1)} {d*p, d*p^2, ..., d*p^e, ...}).

%e 6 is a term since gcd(6, Phi_6(2)) = gcd(6, 3) = 3 > 1.

%e 55 is a term since 55 = 11*5, 5 | (11-1). Indeed, gcd(55, Phi_55(3)) = gcd(55, 8138648440293876241) = 11 > 1.

%e 12 is not a term since 12 = 3*4 but 4 does not divide 3-1. Indeed, gcd(12, Phi_12(a)) = gcd(12, a^4-a^2+1) = 1 for all a.

%o (PARI) isA342256(k) = if(k>1, my(L=factor(k), d=omega(k), p=L[d,1]); (p-1)%(k/p^L[d,2])==0, 0)

%Y Cf. A342255, A342257. Complement of A253235.

%K nonn,easy

%O 1,1

%A _Jianing Song_, Mar 07 2021

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Last modified December 7 18:28 EST 2023. Contains 367660 sequences. (Running on oeis4.)