A175067 a(n) is the sum of perfect divisors of n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 12, 13, 14, 15, 22, 17, 18, 19, 20, 21, 22, 23, 24, 30, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 42, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 56, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 78, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Author

Jaroslav Krizek, Jan 23 2010

Keywords

• nonn

Comments

a(n) > n for perfect powers n = A001597(m) for m > 2.

Examples

			For n = 8: a(8) = 10; there are two perfect divisors of 8: 2 and 8; their sum is 10.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A175070, A007916, A175068, A253641, A052410.

Programs

Maple

A175067:= proc(n) local a, d, k; if n=1 then return 1 end if; a:=0; for d in numtheory[divisors](n) minus {1} do for k do if d^k=n then a:= a+d end if; if n <= d^k then break; end if; end do; end do; a end proc:
seq(A175067(n), n=1..80); # Ridouane Oudra, Dec 12 2024
# second Maple program:
a:= n-> add(`if`(n=d^ilog[d](n), d, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..72);  # Alois P. Heinz, Dec 12 2024

Mathematica

Table[Plus @@ (n^(1/Divisors[GCD @@ FactorInteger[n][[All, 2]]])), {n, 72}] (* Ivan Neretin, May 13 2015 *)

Formula

a(n) = A175070(n) + n. [Jaroslav Krizek, Jan 24 2010]

From Ridouane Oudra, Dec 12 2024: (Start)

a(n) = n, for n in A007916.

a(n^m) = Sum_{d|m} n^d, for n in A007916 and m an integer >0.

More generally, for all integers n we have :

a(n) = Sum_{d|A253641(n)} n^(d/A253641(n)).

a(n) = Sum_{d|A253641(n)} A052410(n)^d. (End)

Extensions

Name edited by Michel Marcus, Jun 13 2018