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

0,5

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

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 *)

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), 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
(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 A358258 A156222

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