

A135767


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


3



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
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OFFSET

1,24


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 nonperiodic subsets of U_n (nth roots of unity) than there are nonperiodic binary words of length n. Otherwise stated, a(n)>0 if there is a nonperiodic subset of U_n with zero sum. Nonperiodic means having no rotational symmetry (except for identity).


LINKS

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


FORMULA

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


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A102466, A102467 ; A001037, A107847.
Sequence in context: A248639 A293959 A333146 * A208575 A355037 A070203
Adjacent sequences: A135764 A135765 A135766 * A135768 A135769 A135770


KEYWORD

easy,nonn


AUTHOR

M. F. Hasler, Jan 14 2008


STATUS

approved



