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A046660 Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity). 45
0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3, 1).

a(n) = 0 for squarefree n.

A162511(n) = (-1)^a(n). - Reinhard Zumkeller, Jul 08 2009

a(n) = the number of divisors of n that are each a composite power of a prime. - Leroy Quet, Dec 02 2009

a(A005117(n)) = 0; a(A060687(n)) = 1; a(A195086(n)) = 2; a(A195087(n)) = 3; a(A195088(n)) = 4; a(A195089(n)) = 5; a(A195090(n)) = 6; a(A195091(n)) = 7; a(A195092(n)) = 8; a(A195093(n)) = 9; a(A195069(n)) = 10. - Reinhard Zumkeller, Nov 29 2015

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.

FORMULA

a(n) = Omega(n) - omega(n) = A001222(n) - A001221(n).

Additive with a(p^e) = e - 1.

a(n) = Sum_{k = 1..A001221(n)} (A124010(n,k) - 1). - Reinhard Zumkeller, Jan 09 2013

G.f.: Sum_{p prime, k>=2} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Jan 06 2017

MAPLE

with(numtheory); A046660:=n->bigomega(n)-nops(factorset(n)); seq(A046660(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013

MATHEMATICA

Table[PrimeOmega[n] - PrimeNu[n], {n, 50}] (* or *) muf[n_] := Module[{fi = FactorInteger[n]}, Total[Transpose[fi][[2]]] - Length[fi]]; Array[muf, 50] (* Harvey P. Dale, Sep 07 2011. The second program is several times faster than the first program for generating large numbers of terms. *)

PROG

(PARI) a(n)=bigomega(n)-omega(n) \\ Charles R Greathouse IV, Nov 14 2012

(PARI) a(n)=my(f=factor(n)[, 2]); vecsum(f)-#f \\ Charles R Greathouse IV, Aug 01 2016

(Haskell)

import Math.NumberTheory.Primes.Factorisation (factorise)

a046660 n = sum es - length es where es = snd $ unzip $ factorise n

-- Reinhard Zumkeller, Nov 28 2015, Jan 09 2013

CROSSREFS

Not the same as A066301.

Cf. A001222, A001221, A257851, A261256, A264959, A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069, A275699.

Sequence in context: A280827 A103840 A066301 * A183094 A108730 A056973

Adjacent sequences:  A046657 A046658 A046659 * A046661 A046662 A046663

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified December 10 20:48 EST 2017. Contains 295856 sequences.