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A342257 Period of the sequence {gcd(n, Phi_n(a)): a in Z}, where Phi_n is the n-th cyclotomic polynomial.

%I A342257 #12 Aug 09 2022 07:05:13
%S A342257 1,2,3,2,5,3,7,2,3,5,11,1,13,7,1,2,17,3,19,5,7,11,23,1,5,13,3,1,29,1,
%T A342257 31,2,1,17,1,1,37,19,13,1,41,7,43,1,1,23,47,1,7,5,1,13,53,3,11,1,19,
%U A342257 29,59,1,61,31,1,2,1,1,67,17,1,1,71,1,73,37,1
%N A342257 Period of the sequence {gcd(n, Phi_n(a)): a in Z}, where Phi_n is the n-th cyclotomic polynomial.
%C A342257 a(n) is the period of the n-th column of A342255. See A342255 for more information.
%C A342257 Also a(n) is the maximum value of the n-th column of A342255. - _Jianing Song_, Aug 09 2022
%H A342257 Jianing Song, <a href="/A342257/b342257.txt">Table of n, a(n) for n = 1..10000</a>
%F A342257 a(n) is the largest prime factor of n if n is in A342256, 1 otherwise.
%e A342257 gcd(6, Phi_6(a)) = gcd(6, a^2-a+1) = 3 for a == 2 (mod 3), 1 otherwise, so {gcd(6, Phi_6(a)): a in Z} has period 3, hence a(6) = 3.
%e A342257 gcd(12, Phi_12(a)) = gcd(12, a^4-a^2+1) = 1 for all n, so {gcd(12, Phi_12(a)): a in Z} has period 1, hence a(12) = 1.
%o A342257 (PARI) a(n) = if(n>1, my(L=factor(n), d=omega(n), p=L[d, 1]); if((p-1)%(n/p^L[d, 2])==0, p, 1), 1)
%Y A342257 Cf. A342255, A253235 (indices of 1), A342256 (indices of terms other than 1), A006530, A013595 (coefficients of cyclotomic polynomials).
%K A342257 nonn,easy
%O A342257 1,2
%A A342257 _Jianing Song_, Mar 07 2021