A168512 Sum of divisors of n weighted by divisor multiplicity in n.

1, 3, 4, 9, 6, 12, 8, 19, 16, 18, 12, 30, 14, 24, 24, 41, 18, 42, 20, 44, 32, 36, 24, 64, 36, 42, 46, 58, 30, 72, 32, 75, 48, 54, 48, 102, 38, 60, 56, 94, 42, 96, 44, 86, 81, 72, 48, 134, 64, 98, 72, 100, 54, 126, 72, 124, 80, 90, 60, 170, 62, 96, 107, 153, 84, 144, 68, 128, 96

Offset: 1

Author

Joseph L. Pe, Nov 28 2009

Keywords

• nonn

Comments

If d > 1 divides n, the multiplicity of d in n is the largest integer i such that d^i divides n; e.g. the multiplicity of 4 in 16 is 2. If d = 1 (degenerate case), then the multiplicity of d is defined as 1.

Examples

			The divisors of 16 are 1, 2, 4, 8, 16, which are of multiplicity 1, 4, 2, 1, 1, respectively, in 16. So a(16) = 1*1 + 4*2 + 2*4 + 1*8 + 1*16 = 41.
		

Links

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

Crossrefs

Cf. A169594, A286561, A286563.

Programs

Mathematica

Table[1 + Total[Function[i, i*Select[Range[Log[i, n]], Divisible[n, i^#] &][[-1]]] /@ Rest@Divisors@n], {n, 69}] (* Ivan Neretin, Jul 26 2015 *)
Table[1 + DivisorSum[n, # IntegerExponent[n, #] &, # > 1 &], {n, 69}] (* Michael De Vlieger, May 20 2017 *)

PARI

A286561(n,k) = { my(i=1); if(1==k, 1, while(!(n%(k^i)), i = i+1); (i-1)); };
A168512(n) = sumdiv(n,d,A286561(n,d)*d); \\ Antti Karttunen, May 20 2017

Formula

a(n) = Sum_{d|n} A286561(n,d)*d. - Antti Karttunen, May 20 2017

Extensions

Extended by Ray Chandler, Dec 08 2009