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A046660
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Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).
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113
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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
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OFFSET
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1,8
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COMMENTS
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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.
a(n) = the number of divisors of n that are each a composite power of a prime. - Leroy Quet, Dec 02 2009
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LINKS
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FORMULA
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Additive with a(p^e) = e - 1.
G.f.: Sum_{p prime, k>=2} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Jan 06 2017
Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141). - Amiram Eldar, Jul 28 2020
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MAPLE
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with(numtheory); A046660 := n -> bigomega(n)-nops(factorset(n)):
# Or:
with(NumberTheory): A046660 := n -> NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jul 14 2023
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MATHEMATICA
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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. *)
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PROG
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(Haskell)
import Math.NumberTheory.Primes.Factorisation (factorise)
a046660 n = sum es - length es where es = snd $ unzip $ factorise n
(Python)
from sympy import factorint
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CROSSREFS
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Cf. A001222, A001221, A003557, A056170, A136141, A257851, A261256, A264959, A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069, A275699, A275812.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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