A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

1, 2, 3, 1, 5, 6, 7, 0, 3, 10, 11, 3, 13, 14, 15, 0, 17, 6, 19, 5, 21, 22, 23, 0, 10, 26, 1, 7, 29, 30, 31, 0, 33, 34, 35, 3, 37, 38, 39, 0, 41, 42, 43, 11, 15, 46, 47, 0, 21, 20, 51, 13, 53, 2, 55, 0, 57, 58, 59, 15, 61, 62, 21, 0, 65, 66

Offset: 1

Author

Peter Luschny, Jan 16 2011

Keywords

• nonn
• mult

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Crossrefs

Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.

Cf. A027748, A124010, A007318, A328745.

Programs

Haskell

a182938 n = product $ zipWith a007318'
   (a027748_row n) (map toInteger $ a124010_row n)
-- Reinhard Zumkeller, Feb 18 2012

Maple

A182938 := proc(n) local e,j; e := ifactors(n)[2]:
mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
seq (A182938(n), n=1..100);

Mathematica

a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
Table[a[n], {n, 1, 100}] (* Kellen Myers, Jan 16 2011 *)

PARI

a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ M. F. Hasler

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ Vaclav Kotesovec, Mar 28 2025

Formula

a(A185359(n)) = 0. - Reinhard Zumkeller, Feb 18 2012

Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - Ilya Gutkovskiy, Oct 26 2019

Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - Vaclav Kotesovec, Mar 28 2025

Extensions

Given terms checked with new PARI code by M. F. Hasler, Jan 16 2011