A318886 -id:A318886 - cp's OEIS frontend

A070536 Number of terms in n-th cyclotomic polynomial minus largest prime factor of n; a(1)=1 by convention.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 10, 0, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 0, 2, 0, 0, 2, 0, 18, 4, 0, 0, 8, 10, 0, 0, 0, 0, 2, 0, 20, 4, 0, 0, 0, 0, 0, 2, 24, 0, 10, 0, 0, 2, 10, 0, 10, 0, 12, 0, 0, 0, 4, 0, 0, 6, 0, 0, 26

Offset: 1

Labos Elemer, May 03 2002

nonn

When (as at n=105) coefficients are not equal 1 or -1 then terms in C[n,x] are counted with multiplicity. - This is the comment by the original author. However, the claim contradicts the given formula, as A051664 counts each nonzero coefficient just once, regardless of its value. For the version summing the absolute values of the coefficients (thus "with multiplicity"), see A318886. - Antti Karttunen, Sep 10 2018

			n=21: Cyclotomic[21,x]=1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12 has 9 terms while largest prime factor of 21 is 7

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A051664, A070537, A070776.

Differs from A318886 for the first time at n=105, where a(105) = 26, while A318886(105) = 28.

Array[Length@ Cyclotomic[#, x] - FactorInteger[#][[-1, 1]] &, 105] (* Michael De Vlieger, Sep 10 2018 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ From A051664
A070536(n) = (A051664(n) - A006530(n)); \\ Antti Karttunen, Sep 10 2018

a(n) = A051664(n) - A006530(n).

Data section extended to 105 terms by Antti Karttunen, Sep 10 2018

A318884 a(n) is the sum of absolute values of the coefficients in the n-th cyclotomic polynomial.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3, 41, 83, 9, 41, 43, 39, 11, 89

Offset: 1

Antti Karttunen, Sep 10 2018

nonn

Differs from A051664 in the positions given by A013590, thus for the first time at n=105, where a(105) = 35, while A051664(105) = 33 as the 105th cyclotomic polynomial is the first one that has a coefficient other than 1, 0, or -1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Cyclotomic polynomial

Cf. A013590, A013595, A051258, A051664, A318886.

Array[Total@ Abs@ CoefficientList[Cyclotomic[#, x], x] &, 89] (* Michael De Vlieger, Sep 10 2018 *)

A318884(n) = vecsum(apply(abs,Vec(polcyclo(n)))); \\ Antti Karttunen, Sep 10 2018

cp's OEIS Frontend

A070536 Number of terms in n-th cyclotomic polynomial minus largest prime factor of n; a(1)=1 by convention.

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