A070536 - OEIS

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

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OFFSET

1,15

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.

A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ From A051664