A072481 a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d).

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

Offset: 0

Author

Reinhard Zumkeller, Aug 02 2002

Keywords

• nonn

Comments

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

Partial sums of A004125.

Links

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

Crossrefs

Cf. A224923, A224924.

Programs

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

Mathematica

a[n_] := n(n+1)(2n+1)/6 - Sum[DivisorSigma[1, k] (n-k+1), {k, 1, n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 08 2019, after Omar E. Pol *)

PARI

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

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), end=',')

Python

from math import isqrt
def A072481(n): return (n*(n+1)*((n<<1)+1)-((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1)-sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 22 2023

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

Extensions

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

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