A064478 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(0) = 1, a(1)=2.

1, 2, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144

Offset: 0

Author

N. J. A. Sloane, Oct 06 2001

Keywords

• nonn
• nice
• easy

Comments

a(0)=1 and a(1)=2 by convention (which makes a(n) not multiplicative).

The alternate convention a(0)=0 and a(1)=1 would have made a(n) completely multiplicative (cf. A003959 for completely multiplicative version.) - Daniel Forgues, Nov 17 2009

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A064476, A064479, A003958. Apart from initial terms, same as A003959.

Programs

Haskell

a064478 n = if n <= 1 then n + 1 else a003959 n
-- Reinhard Zumkeller, Feb 28 2013

Maple

a:= n-> `if`(n<2, n+1, mul((i[1]+1)^i[2], i=ifactors(n)[2])):
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 13 2017

Mathematica

a[0] = 1; a[1] = 2; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); Table[a[n], {n, 0, 66}](* Jean-François Alcover, Nov 14 2011 *)
f[n_] := Times @@ ((1 + #[[1]])^#[[2]] & /@ FactorInteger@ n); Array[f, 67, 0] (* Robert G. Wilson v, Sep 13 2017 *)

PARI

a(n) = if(n<=1, n + 1, my(f=factor(n)); prod(i=1, #f~, (1 + f[i, 1])^f[i, 2])) \\ Harry J. Smith, Sep 15 2009

Python

from math import prod
from sympy import factorint
def A064478(n): return prod((p+1)**e for p,e in factorint(n).items()) if n!=1 else 2 # Chai Wah Wu, Mar 26 2025

Extensions

More terms from Vladeta Jovovic, Oct 06 2001

Edited by Daniel Forgues, Nov 18 2009