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A008481 If n = Product (p_j^k_j) then a(n) = Sum partition(k_j).

%I A008481 #32 Sep 29 2023 04:05:08
%S A008481 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,5,1,3,1,3,2,2,1,4,2,2,3,3,1,3,1,7,2,2,
%T A008481 2,4,1,2,2,4,1,3,1,3,3,2,1,6,2,3,2,3,1,4,2,4,2,2,1,4,1,2,3,11,2,3,1,3,
%U A008481 2,3,1,5,1,2,3,3,2,3,1,6,5,2,1,4,2,2,2
%N A008481 If n = Product (p_j^k_j) then a(n) = Sum partition(k_j).
%C A008481 a(n) is a function of the prime signature of n (cf. A025487). - _Matthew Vandermast_, Jun 24 2012
%H A008481 Antti Karttunen, <a href="/A008481/b008481.txt">Table of n, a(n) for n = 1..65537</a> (first 10000 terms from Vincenzo Librandi)
%H A008481 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A008481 From _Antti Karttunen_, Aug 30 2018: (Start)
%F A008481 Additive with a(p^e) = A000041(e).
%F A008481 a(n) = A007814(A318312(n)). (End)
%F A008481 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 1.03089282973521424158..., where f(x) = -1 + (1-x) * Product_{k>=1} (1 + x^k)/(1 - x^(2*k)). - _Amiram Eldar_, Sep 29 2023
%p A008481 a:= n-> add(combinat[numbpart](i[2]), i=ifactors(n)[2]):
%p A008481 seq(a(n), n=1..100);  # _Alois P. Heinz_, Aug 30 2018
%t A008481 Prepend[ Array[ Plus @@ (PartitionsP /@ Last[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 0 ]
%t A008481 (* Second program: *)
%t A008481 Array[Total[PartitionsP /@ FactorInteger[#][[All, -1]] - Boole[# == 1]] &, 87] (* _Michael De Vlieger_, Sep 02 2018 *)
%o A008481 (PARI) A008481(n) = vecsum(apply(e -> numbpart(e),factor(n)[,2])); \\ _Antti Karttunen_, Aug 30 2018
%Y A008481 Cf. A000041, A007814, A025487, A077761, A318312.
%Y A008481 Differs from A318473 for the first time at n=32, where a(32)=7, while A318473(32)=8.
%K A008481 nonn,easy
%O A008481 1,4
%A A008481 _Olivier Gérard_
%E A008481 Term a(1) corrected from 1 to 0 (for an empty sum) by _Antti Karttunen_, Aug 30 2018