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%I #61 Oct 21 2023 17:07:00
%S 1,7,19,47,77,149,205,325,442,622,754,1090,1272,1608,1968,2464,2770,
%T 3472,3852,4692,5364,6156,6708,8148,8923,10015,11095,12663,13533,
%U 15693,16685,18701,20285,22121,23801,27077,28483,30763,32947,36547
%N a(n) = Sum_{k=1..n} k*sigma(k).
%C Partial sums of A064987. - _Omar E. Pol_, Jul 04 2014
%C a(n) is also the volume after n-th step of the symmetric staircase described in A244580 (see also A237593). - _Omar E. Pol_, Jul 31 2018
%C In general, for j >= 1 and m >= 0, Sum_{k=1..n} k^m * sigma_j(k) ~ n^(j+m+1) * zeta(j+1) / (j+m+1). - _Daniel Suteu_, Nov 21 2018
%H Indranil Ghosh, <a href="/A143128/b143128.txt">Table of n, a(n) for n = 1..4267</a>
%F Sum {k=1..n} k*sigma(k), where sigma(n) = A000203: (1, 3, 4, 7, 6, 12, ...) and n*sigma(n) = A064987: (1, 6, 12, 28, ...). Equals row sums of triangle A110662. - _Emeric Deutsch_, Aug 12 2008
%F a(n) ~ n^3 * Pi^2/18. - _Charles R Greathouse IV_, Jun 19 2012
%F G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 13 2017
%F a(n) = Sum_{k=1..n} k^2/2 * floor(n/k) * floor(1 + n/k). - _Daniel Suteu_, May 29 2018
%F a(n) = A256533(n) - A175254(n-1), n >= 2. - _Omar E. Pol_, Jul 31 2018
%F a(n) = Sum_{k=1..s} (k*A000330(floor(n/k)) + k^2*A000217(floor(n/k))) - A000330(s)*A000217(s), where s = floor(sqrt(n)). - _Daniel Suteu_, Nov 26 2020
%F a(n) = Sum_{k=1..n} Sum_{i=1..floor(n/k)} i*k^2. - _Wesley Ivan Hurt_, Nov 26 2020
%e a(4) = 47 = (1 + 6 + 12 + 28) where A064987 = (1, 6, 12, 28, 30, ...).
%e a(4) = 47 = sum of row 4 terms of triangle A110662 = (15 + 14 + 11 + 7).
%p with(numtheory): a:=proc(n) options operator, arrow: sum(k*sigma(k), k=1..n) end proc: seq(a(n),n=1..40); # _Emeric Deutsch_, Aug 12 2008
%t Table[Sum[i*DivisorSigma[1, i], {i, n}], {n, 50}] (* _Wesley Ivan Hurt_, Jul 06 2014 *)
%o (PARI) a(n)=sum(k=1,n,k*sigma(k)) \\ _Charles R Greathouse IV_, Apr 27 2015
%o (PARI) f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
%o g(n) = n*(n+1)/2; \\ A000217
%o a(n) = sum(k=1, sqrtint(n), k * f(n\k) + k^2 * g(n\k)) - f(sqrtint(n)) * g(sqrtint(n)); \\ _Daniel Suteu_, Nov 26 2020
%o (Magma) [(&+[k*DivisorSigma(1,k): k in [1..n]]): n in [1..50]]; // _G. C. Greubel_, Nov 21 2018
%o (Sage) [sum(k*sigma(k,1) for k in (1..n)) for n in (1..50)] # _G. C. Greubel_, Nov 21 2018
%o (Python)
%o def A143128(n): return sum(k**2*(m:=n//k)*(m+1)>>1 for k in range(1,n+1)) # _Chai Wah Wu_, Oct 20 2023
%o (Python)
%o from math import isqrt
%o def A143128(n): return ((-((s:=isqrt(n))*(s+1))**2*(2*s+1)>>1) + sum((q:=n//k)*(q+1)*k*(3*k+2*q+1) for k in range(1,s+1)))//6 # _Chai Wah Wu_, Oct 21 2023
%Y Cf. A000203, A064987, A110662, A175254, A237593, A244580, A256533.
%K nonn
%O 1,2
%A _Gary W. Adamson_, Jul 26 2008
%E Corrected and extended by _Emeric Deutsch_, Aug 12 2008
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