OFFSET
1,4
COMMENTS
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).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000 (the first 10000 terms from Reinhard Zumkeller)
FORMULA
Dirichlet g.f.: Product_{n in A050376} (1/(1-1/n^s)).
a(p^k) = A000123([k/2]) for all primes p.
a(A002110(n)) = 1.
a(n) = Sum{a(d): d^2 divides n}, a(1) = 1. - Reinhard Zumkeller, Jul 12 2007
G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Nov 25 2020
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.7876368001694456669..., where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)). - Amiram Eldar, Oct 03 2023
MAPLE
A018819:= proc(n) option remember;
if n::odd then procname(n-1)
else procname(n-1) + procname(n/2)
fi
end proc:
A018819(0):= 1:
f:= n -> mul(A018819(s[2]), s=ifactors(n)[2]):
seq(f(n), n=1..100); # Robert Israel, Jan 14 2016
MATHEMATICA
b[0] = 1; b[n_] := b[n] = b[n - 1] + If[EvenQ[n], b[n/2], 0];
a[n_] := Times @@ (b /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 27 2018 *)
PROG
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Christian G. Bower, Nov 15 1999
EXTENSIONS
More terms from Antti Karttunen, Dec 28 2019
STATUS
approved