A396283 a(n) = Sum_{k=1..n} (n mod k)^2.

0, 0, 1, 1, 6, 5, 16, 18, 32, 35, 70, 59, 104, 119, 160, 162, 249, 237, 348, 349, 432, 467, 642, 587, 755, 812, 971, 982, 1257, 1188, 1493, 1517, 1752, 1859, 2212, 2068, 2499, 2636, 2991, 2958, 3513, 3412, 4021, 4088, 4447, 4636, 5397, 5144, 5890, 6012, 6603, 6686, 7645, 7524, 8415

Offset

1,5

Links

Table of n, a(n) for n=1..55.

Formula

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

a(n) = n * ( A000290(n) - 2 * A024916(n) ) + A350123(n).

a(n) ~ n^3 * (1 - Pi^2/18 - zeta(3)/3).

Mathematica

Table[Sum[Mod[n, k]^2, {k, n}], {n, 55}]

Prog

(Python)
def A396283(n):
    c, j = n**3, 1
    while j <= n:
        k = n//j
        m = n//k
        c += k**2*((a:=m*(m+1))*(2*m+1)-(b:=(j-1)*j)*(2*j-1))//6-n*k*(a-b)
        j = m+1
    return c # Chai Wah Wu, May 21 2026

Crossrefs

Cf. A000290, A004125, A024916, A051126, A067459, A309176, A350123, A396284.

Sequence in context: A300750 A101493 A347276 * A188067 A039668 A281175

Adjacent sequences: A396279 A396281 A396282 * A396284 A396285 A396286

Keyword

nonn

Author

Ilya Gutkovskiy, May 21 2026

Status

approved