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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). 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 (list; graph; refs; listen; history; text; internal format)
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 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

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

PROG

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

CROSSREFS

Cf. A102466, A102467 ; A001037, A107847.

Sequence in context: A131371 A003475 A248639 * A208575 A070203 A070201

Adjacent sequences:  A135764 A135765 A135766 * A135768 A135769 A135770

KEYWORD

easy,nonn

AUTHOR

M. F. Hasler, Jan 14 2008

STATUS

approved

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Last modified May 29 16:53 EDT 2017. Contains 287252 sequences.