A396284 a(n) = Sum_{k=1..n} (n mod k)^3.

0, 0, 1, 1, 10, 9, 38, 44, 102, 109, 262, 233, 472, 529, 840, 858, 1467, 1403, 2272, 2329, 3208, 3395, 5048, 4735, 6649, 7042, 9083, 9238, 12583, 12084, 16091, 16421, 20146, 21065, 26798, 25518, 32351, 33716, 40181, 40290, 49977, 48848, 59977, 61006, 70041, 72268, 87295, 84266, 100524, 102726

Offset

1,5

Links

Table of n, a(n) for n=1..50.

Formula

a(n) = n^4 - Sum_{k=1..n} k * floor(n/k) * ( 3 * n^2 - 3 * n * k * floor(n/k) + ( k * floor(n/k) )^2 ).

a(n) = n * ( A000578(n) - 3 * ( n * A024916(n) - A350123(n) ) ) - A356249(n).

a(n) ~ n^4 * (1 - Pi^2/24 - Pi^4/360 - zeta(3)/4).

Mathematica

Table[Sum[Mod[n, k]^3, {k, n}], {n, 50}]

Prog

(Python)
def A396284(n):
    c, j = n**4, 1
    while j <= n:
        k = n//j
        m = n//k
        c += (n*k**2*((a:=m*(m+1))*(2*m+1)-(b:=(j-1)*j)*(2*j-1))-3*n**2*k*(e:=a-b)>>1)-(k**3*e*(a+b)>>2)
        j = m+1
    return c # Chai Wah Wu, May 21 2026

Crossrefs

Cf. A000578, A004125, A024916, A051126, A309176, A350123, A356249, A396283.

Sequence in context: A271973 A237114 A217412 * A241285 A052196 A243035

Adjacent sequences: A396281 A396282 A396283 * A396285 A396286 A396287

Keyword

nonn

Author

Ilya Gutkovskiy, May 21 2026

Status

approved