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A168512 Sum of divisors of n weighted by divisor multiplicity in n. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A319021 A249187 A084425 * A003959 A168341 A083111

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

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Last modified June 25 11:56 EDT 2019. Contains 324352 sequences. (Running on oeis4.)