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%I #63 Sep 28 2023 06:46:58
%S 1,6,16,37,63,113,163,248,339,469,591,801,971,1221,1481,1822,2112,
%T 2567,2929,3475,3975,4585,5115,5965,6616,7466,8286,9336,10178,11478,
%U 12440,13805,15025,16475,17775,19686,21056,22866,24566,26776,28458,30958
%N Partial sums of A001157: Sum_{j=1..n} sigma_2(j).
%C In general, for m >= 0 and j >= 0, Sum_{k=1..n} k^m * sigma_j(k) = Sum_{k=1..s} (k^m * F_{m+j}(floor(n/k)) + k^(m+j) * F_m(floor(n/k))) - F_{m+j}(s) * F_m(s), where s = floor(sqrt(n)) and F_m(x) are the Faulhaber polynomials defined as F_m(x) = (Bernoulli(m+1, x+1) - Bernoulli(m+1, 1)) / (m+1). - _Daniel Suteu_, Nov 27 2020
%H Seiichi Manyama, <a href="/A064602/b064602.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H Project Euler, <a href="http://projecteuler.net/problem=401">Problem 401: Sum of squares of divisors</a>, 2012.
%F a(n) = a(n-1) + A001157(n) = Sum_{j=1..n} sigma_2(j) where sigma_2(j) = A001157(j).
%F a(n) = Sum_{i=1..n} i^2 * floor(n/i). - _Enrique Pérez Herrero_, Sep 15 2012
%F G.f.: (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 02 2017
%F a(n) ~ zeta(3) * n^3 / 3. - _Vaclav Kotesovec_, Sep 02 2018
%F a(n) = Sum_{k=1..s} (A000330(floor(n/k)) + k^2*floor(n/k)) - s*A000330(s), where s = floor(sqrt(n)). - _Daniel Suteu_, Nov 26 2020
%t Accumulate@ Array[DivisorSigma[2, #] &, 42] (* _Michael De Vlieger_, Jan 02 2017 *)
%o (PARI) a(n) = sum(j=1, n, sigma(j, 2)); \\ _Michel Marcus_, Dec 15 2013
%o (PARI) f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
%o a(n) = my(s=sqrtint(n)); sum(k=1, s, f(n\k) + k^2*(n\k)) - s*f(s); \\ _Daniel Suteu_, Nov 26 2020
%o (Python)
%o from math import isqrt
%o def f(n): return n*(n+1)*(2*n+1)//6
%o def a(n):
%o s = isqrt(n)
%o return sum(f(n//k) + k*k*(n//k) for k in range(1, s+1)) - s*f(s)
%o print([a(k) for k in range(1, 43)]) # _Michael S. Branicky_, Oct 01 2022 after _Daniel Suteu_
%Y Cf. A001157, A064605.
%Y Cf. A064603, A064604, A248076.
%K nonn
%O 1,2
%A _Labos Elemer_, Sep 24 2001
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