A135767 sigma_0(n)-omega(n)-Omega(n) (sigma_0 = A000005 = # divisors, omega = A001221 = # prime factors, Omega = A001222 = # prime factors with multiplicity).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 5, 0, 0, 1, 1, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 2, 0, 5, 0, 1, 0, 0, 0, 4, 0, 1, 1, 3, 0, 2, 0, 2, 2

Offset: 1

Author

M. F. Hasler, Jan 14 2008

Keywords

• easy
• nonn

Comments

A102467 = { n | a(n)>0 } ; A102466 = { n | a(n)=0 } = { n | omega(n)=1 or Omega(n)=2 }: these are exactly the prime powers (>1) and semiprimes. For all other numbers a(n) > 0 since for each of the Omega(n) prime power divisors, other divisors are obtained by multiplying it with another prime factor, which gives more than omega(n) different additional divisors. a(n)>0 is also equivalent to A001037(n) > A107847(n), i.e. there are strictly fewer nonzero sums of non-periodic subsets of U_n (n-th roots of unity) than there are non-periodic binary words of length n. Otherwise stated, a(n)>0 if there is a non-periodic subset of U_n with zero sum. Non-periodic means having no rotational symmetry (except for identity).

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A102466, A102467 ; A001037, A107847.

Programs

Mathematica

a[n_] := DivisorSigma[0, n] - PrimeOmega[n] - PrimeNu[n];
Array[a, 105] (* Jean-François Alcover, Jun 21 2018 *)

PARI

A135767(n)=numdiv(n)-omega(n)-bigomega(n)

Formula

a(n)=0 <=> omega(n)=1 or Omega(n)=2 <=> n is semiprime or a prime power (>1) <=> A001037(n) = A107847(n) <=> all non-periodic subsets of U_n have nonzero sum