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

%I A135767 #7 Jun 21 2018 21:22:28
%S A135767 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,
%T A135767 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,
%U A135767 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
%N A135767 sigma_0(n)-omega(n)-Omega(n) (sigma_0 = A000005 = # divisors, omega = A001221 = # prime factors, Omega = A001222 = # prime factors with multiplicity).
%C A135767 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).
%H A135767 M. F. Hasler, <a href="/A135767/b135767.txt">Table of n, a(n) for n = 1..10000</a>
%F A135767 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
%t A135767 a[n_] := DivisorSigma[0, n] - PrimeOmega[n] - PrimeNu[n];
%t A135767 Array[a, 105] (* _Jean-François Alcover_, Jun 21 2018 *)
%o A135767 (PARI) A135767(n)=numdiv(n)-omega(n)-bigomega(n)
%Y A135767 Cf. A102466, A102467 ; A001037, A107847.
%K A135767 easy,nonn
%O A135767 1,24
%A A135767 _M. F. Hasler_, Jan 14 2008