|
%I #45 Oct 21 2023 20:02:59
%S 1,10,38,111,237,489,833,1418,2175,3309,4641,6685,8883,11979,15507,
%T 20188,25102,31915,38775,47973,57605,69593,81761,98141,113892,133674,
%U 154114,179226,203616,235368,265160,302609,339905,384131,427475,482736
%N Partial sums of A001158: Sum_{j=1..n} sigma_3(j).
%C In general, Sum_{k=1..n} sigma_m(k) = Sum_{k=1..n} k^m * floor(n/k). - _Daniel Suteu_, Nov 08 2018
%H Seiichi Manyama, <a href="/A064603/b064603.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%F a(n) = a(n-1) + A001158(n) = Sum_{j=1..n} sigma_3(j), where sigma_3(j) = A001158(j).
%F G.f.: (1/(1 - x))*Sum_{k>=1} k^3*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 23 2017
%F a(n) ~ Pi^4 * n^4 / 360. - _Vaclav Kotesovec_, Sep 02 2018
%F a(n) = Sum_{k=1..n} ((1/2) * floor(n/k) * floor(1 + n/k))^2. - _Daniel Suteu_, Nov 07 2018
%F a(n) = Sum_{k=1..n} k^3 * floor(n/k). - _Daniel Suteu_, Nov 08 2018
%t Accumulate@ Array[DivisorSigma[3, #] &, 36] (* _Michael De Vlieger_, Nov 03 2017 *)
%o (PARI) a(n) = sum(j=1, n, sigma(j, 3)); \\ _Michel Marcus_, Nov 04 2017
%o (PARI) a(n) = sum(k=1, n, k^3 * (n\k)); \\ _Daniel Suteu_, Nov 08 2018
%o (Python)
%o from math import isqrt
%o def A064603(n): return (-(s:=isqrt(n))**3*(s+1)**2 + sum((q:=n//k)*(4*k**3+q*(q*(q+2)+1)) for k in range(1,s+1)))>>2 # _Chai Wah Wu_, Oct 21 2023
%Y Cf. A001158, A064606.
%Y Cf. A064602, A064604, A248076.
%K nonn
%O 1,2
%A _Labos Elemer_, Sep 24 2001
|
