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%I #78 Feb 06 2024 20:16:31
%S 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,
%T 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,
%U 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
%N Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).
%C 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).
%C a(n) = 0 for squarefree n.
%C A162511(n) = (-1)^a(n). - _Reinhard Zumkeller_, Jul 08 2009
%C a(n) = the number of divisors of n that are each a composite power of a prime. - _Leroy Quet_, Dec 02 2009
%C 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
%H Charles R Greathouse IV, <a href="/A046660/b046660.txt">Table of n, a(n) for n = 1..10000</a>
%H Mark Kac, <a href="https://web.archive.org/web/20220609160228/http://www.gibbs.if.usp.br/~marchett/estocastica/MarkKac-Statistical-Independence.pdf">Statistical Independence in Probability, Analysis and Number Theory</a>, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
%F a(n) = Omega(n) - omega(n) = A001222(n) - A001221(n).
%F Additive with a(p^e) = e - 1.
%F a(n) = Sum_{k = 1..A001221(n)} (A124010(n,k) - 1). - _Reinhard Zumkeller_, Jan 09 2013
%F G.f.: Sum_{p prime, k>=2} x^(p^k)/(1 - x^(p^k)). - _Ilya Gutkovskiy_, Jan 06 2017
%F 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
%F a(n) = Sum_{p|n} A286563(n/p,p), where p is prime. - _Ridouane Oudra_, Sep 13 2023
%F a(n) = A275812(n) - A056170(n). - _Amiram Eldar_, Jan 09 2024
%F a(n) = A001222(A003557(n)). - _Peter Munn_, Feb 06 2024
%p with(numtheory); A046660 := n -> bigomega(n)-nops(factorset(n)):
%p seq(A046660(k), k=1..100); # _Wesley Ivan Hurt_, Oct 27 2013
%p # Or:
%p with(NumberTheory): A046660 := n -> NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct'): # _Peter Luschny_, Jul 14 2023
%t 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. *)
%o (PARI) a(n)=bigomega(n)-omega(n) \\ _Charles R Greathouse IV_, Nov 14 2012
%o (PARI) a(n)=my(f=factor(n)[,2]); vecsum(f)-#f \\ _Charles R Greathouse IV_, Aug 01 2016
%o (Haskell)
%o import Math.NumberTheory.Primes.Factorisation (factorise)
%o a046660 n = sum es - length es where es = snd $ unzip $ factorise n
%o -- _Reinhard Zumkeller_, Nov 28 2015, Jan 09 2013
%o (Python)
%o from sympy import factorint
%o def A046660(n): return sum(e-1 for e in factorint(n).values()) # _Chai Wah Wu_, Jul 18 2023
%Y Not the same as A066301.
%Y Cf. A001222, A001221, A003557, A056170, A136141, A257851, A261256, A264959, A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069, A275699, A275812.
%K nonn,easy,nice
%O 1,8
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_
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