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

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, 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, 29, 59, 1, 61, 31, 1, 2, 1, 1, 67, 17, 1, 1, 71, 1, 73, 37, 1

Offset

1,2

Comments

a(n) is the period of the n-th column of A342255. See A342255 for more information.

Also a(n) is the maximum value of the n-th column of A342255. - Jianing Song, Aug 09 2022

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Formula

a(n) is the largest prime factor of n if n is in A342256, 1 otherwise.

Example

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.
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.

Prog

(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)

Crossrefs

Cf. A342255, A253235 (indices of 1), A342256 (indices of terms other than 1), A006530, A013595 (coefficients of cyclotomic polynomials).

Sequence in context: A262549 A086287 A253236 * A286516 A273289 A365521

Adjacent sequences: A342254 A342255 A342256 * A342258 A342259 A342260

Keyword

nonn,easy

Author

Jianing Song, Mar 07 2021

Status

approved