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A072481 a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d). 3
0, 0, 0, 1, 2, 6, 9, 17, 25, 37, 50, 72, 89, 117, 148, 184, 220, 271, 318, 382, 443, 513, 590, 688, 773, 876, 988, 1113, 1237, 1388, 1526, 1693, 1860, 2044, 2241, 2459, 2657, 2890, 3138, 3407, 3665, 3962, 4246, 4571, 4899, 5238, 5596, 5999, 6373, 6787, 7207 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Previous name was: Sums of sums of remainders when dividing n by k, 0<k<=n.

Partial sums of A004125.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = Sum_{k=1..n}Sum_{d=1..k}(k mod d).

a(n) = A000330(n) - A175254(n), n >= 1. - Omar E. Pol, Aug 12 2015

G.f.: x^2/(1-x)^4 - (1-x)^(-2) * Sum_{k>=1} k*x^(2*k)/(1-x^k). - Robert Israel, Aug 13 2015

a(n) ~ (1 - Pi^2/12)*n^3/3. - Vaclav Kotesovec, Sep 25 2016

MAPLE

N:= 200: # to get a(0) to a(N)

S:= series(add(k*x^(2*k)/(1-x^k), k=1..floor(N/2))/(1-x)^2, x, N+1):

seq((n^3-n)/6 - coeff(S, x, n), n=0..N); # Robert Israel, Aug 13 2015

PROG

(Python)

for n in range(99):

    s = 0

    for k in range(1, n+1):

      for d in range(1, k+1):

        s += k % d

    print str(s)+', ',

(PARI) a(n) = sum(k=1, n, sum(d=1, k, k % d)); \\ Michel Marcus, Feb 11 2014

CROSSREFS

Cf. A224923, A224924.

Sequence in context: A254057 A257083 A054974 * A032471 A156222 A002886

Adjacent sequences:  A072478 A072479 A072480 * A072482 A072483 A072484

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Aug 02 2002

EXTENSIONS

New name and a(0) from Alex Ratushnyak, Feb 10 2014

STATUS

approved

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Last modified May 30 03:33 EDT 2017. Contains 287305 sequences.