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A046660
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Excess of n = number of primes divisors (with multiplicity) - number of prime divisors (without multiplicity).
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23
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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.
A162511(n) = (-1)^a(n). [From Reinhard Zumkeller, Jul 08 2009]
a(n) = the number of divisors of n that are each a composite power of a prime. [From Leroy Quet, Dec 02 2009]
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REFERENCES
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M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = Omega(n) - omega(n) = A001222(n) - A001221(n).
Additive with a(p^e) = e - 1.
a(n) = sum(A124010(n,k)-1: k=1..A001221(n)). - Reinhard Zumkeller, Jan 09 2013
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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](* From 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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(PARI) a(n)=bigomega(n)-omega(n) \\ Charles R Greathouse IV, Nov 14 2012
(Haskell)
a046660 1 = 0
a046660 n = sum $ map (subtract 1) $ a124010_row n
-- Reinhard Zumkeller, Jan 09 2013
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CROSSREFS
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Cf. A001222, A001221. Not the same as A066301.
Sequence in context: A081221 A103840 A066301 * A183094 A108730 A056973
Adjacent sequences: A046657 A046658 A046659 * A046661 A046662 A046663
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from David W. Wilson.
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STATUS
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approved
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