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A008481
If n = Product (p_j^k_j) then a(n) = Sum partition(k_j).
5
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, 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, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 6, 5, 2, 1, 4, 2, 2, 2
OFFSET
1,4
COMMENTS
a(n) is a function of the prime signature of n (cf. A025487). - Matthew Vandermast, Jun 24 2012
FORMULA
From Antti Karttunen, Aug 30 2018: (Start)
Additive with a(p^e) = A000041(e).
a(n) = A007814(A318312(n)). (End)
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
MAPLE
a:= n-> add(combinat[numbpart](i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 30 2018
MATHEMATICA
Prepend[ Array[ Plus @@ (PartitionsP /@ Last[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 0 ]
(* Second program: *)
Array[Total[PartitionsP /@ FactorInteger[#][[All, -1]] - Boole[# == 1]] &, 87] (* Michael De Vlieger, Sep 02 2018 *)
PROG
(PARI) A008481(n) = vecsum(apply(e -> numbpart(e), factor(n)[, 2])); \\ Antti Karttunen, Aug 30 2018
CROSSREFS
Differs from A318473 for the first time at n=32, where a(32)=7, while A318473(32)=8.
Sequence in context: A069248 A329378 A329617 * A318473 A127669 A323436
KEYWORD
nonn,easy
EXTENSIONS
Term a(1) corrected from 1 to 0 (for an empty sum) by Antti Karttunen, Aug 30 2018
STATUS
approved