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

1,2

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.

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A169594, A286561, A286563.

Sequence in context: A249187 A084425 A365480 * A337180 A003959 A344461

Adjacent sequences: A168509 A168510 A168511 * A168513 A168514 A168515

Keyword

nonn

Author

Joseph L. Pe, Nov 28 2009

Extensions

Extended by Ray Chandler, Dec 08 2009

Status

approved