A126164 Sum of the proper exponential divisors of n.

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 10, 0, 0, 0, 6, 5, 0, 3, 14, 0, 0, 0, 2, 0, 0, 0, 36, 0, 0, 0, 10, 0, 0, 0, 22, 15, 0, 0, 18, 7, 10, 0, 26, 0, 6, 0, 14, 0, 0, 0, 30, 0, 0, 21, 14, 0, 0, 0, 34, 0, 0, 0, 48, 0, 0, 15

Offset: 1

Author

Ant King, Dec 21 2006

Keywords

• easy
• nonn

Comments

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

Examples

			The exponential divisors of 240 are 30, 60 and 240, so a(240) = 30+60 = 90.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Ant King, Mathematica programs for A126164 - A126166.
• Eric Weisstein's World of Mathematics, e-Divisor.

Crossrefs

Cf. A051377, A049419, A054979, A054980, A126168.

Programs

Mathematica

f[p_, e_] := DivisorSum[e, p^# &]; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

PARI

A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ This function from Charles R Greathouse IV, Nov 22 2011
A126164(n) = (A051377(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula

Formula

a(n) = esigma(n) - n = A051377(n) - n.