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

%I A064478 #32 Mar 26 2025 20:23:13
%S A064478 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,
%T A064478 36,42,64,72,30,72,32,243,48,54,48,144,38,60,56,162,42,96,44,108,96,
%U A064478 72,48,324,64,108,72,126,54,192,72,216,80,90,60,216,62,96,128,729,84,144
%N A064478 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(0) = 1, a(1)=2.
%C A064478 a(0)=1 and a(1)=2 by convention (which makes a(n) not multiplicative).
%C A064478 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
%H A064478 Harry J. Smith, <a href="/A064478/b064478.txt">Table of n, a(n) for n = 0..1000</a>
%p A064478 a:= n-> `if`(n<2, n+1, mul((i[1]+1)^i[2], i=ifactors(n)[2])):
%p A064478 seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 13 2017
%t A064478 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 *)
%t A064478 f[n_] := Times @@ ((1 + #[[1]])^#[[2]] & /@ FactorInteger@ n); Array[f, 67, 0] (* _Robert G. Wilson v_, Sep 13 2017 *)
%o A064478 (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
%o A064478 (Haskell)
%o A064478 a064478 n = if n <= 1 then n + 1 else a003959 n
%o A064478 -- _Reinhard Zumkeller_, Feb 28 2013
%o A064478 (Python)
%o A064478 from math import prod
%o A064478 from sympy import factorint
%o A064478 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
%Y A064478 Cf. A064476, A064479, A003958. Apart from initial terms, same as A003959.
%K A064478 nonn,nice,easy
%O A064478 0,2
%A A064478 _N. J. A. Sloane_, Oct 06 2001
%E A064478 More terms from _Vladeta Jovovic_, Oct 06 2001
%E A064478 Edited by _Daniel Forgues_, Nov 18 2009