A356043 a(n) = Sum_{k=1..n} sigma_3(k) * floor(n/k).

1, 11, 40, 123, 250, 540, 885, 1553, 2339, 3609, 4942, 7349, 9548, 12998, 16681, 22030, 26945, 34805, 41666, 52207, 62212, 75542, 87711, 107083, 122961, 144951, 166177, 194812, 219203, 256033, 285826, 328624, 367281, 416431, 460246, 525484, 576139, 644749, 708520

Offset

1,2

Links

Table of n, a(n) for n=1..39.

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} d^3 * tau(k/d).

G.f.: (1/(1-x)) * Sum_{k>=1} sigma_3(k) * x^k/(1 - x^k).

a(n) ~ Pi^8 * n^4 / 32400. - Vaclav Kotesovec, Aug 07 2022

Mathematica

Table[Sum[DivisorSigma[3, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 07 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, sigma(k, 3)*(n\k));
(PARI) a(n) = sum(k=1, n, sumdiv(k, d, d^3*numdiv(k/d)));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 3)*x^k/(1-x^k))/(1-x))
(Python)
from math import isqrt
def A356043(n):
    c, j, v = 0, 1, 0
    while j <= n:
        k = n//j
        m = n//k
        c += k*(-v+(v:=-(s:=isqrt(m))**3*(s+1)**2+sum((q:=m//a)*(4*a**3+q*(q*(q+2)+1)) for a in range(1, s+1))))
        j = m+1
    return c>>2 # Chai Wah Wu, May 21 2026

Crossrefs

Partial sums of A321140.

Column k=3 of A356045.

Cf. A000005 (tau), A064603.

Sequence in context: A059142 A064798 A056124 * A225919 A348586 A064768

Adjacent sequences: A356040 A356041 A356042 * A356044 A356045 A356046

Keyword

nonn

Author

Seiichi Manyama, Jul 24 2022

Status

approved