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%I #85 Oct 22 2023 00:38:39
%S 1,5,13,28,49,82,123,179,248,335,434,561,702,867,1056,1276,1514,1791,
%T 2088,2427,2798,3205,3636,4127,4649,5213,5817,6477,7167,7929,8723,
%U 9580,10485,11444,12451,13549,14685,15881,17133,18475,19859,21339,22863,24471,26157
%N a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.
%C Partial sums of A024916. - _Omar E. Pol_, Jul 03 2014
%C a(n) is also the volume of the stepped pyramid with n levels described in A245092. - _Omar E. Pol_, Aug 12 2015
%C Also the alternating row sums of A262612. - _Omar E. Pol_, Nov 23 2015
%C From _Omar E. Pol_, Jan 20 2021: (Start)
%C Convolution of A000203 and A000027.
%C Convolution of A340793 and the nonzero terms of A000217.
%C Antidiagonal sums of A319073.
%C Row sums of A274824. (End)
%C Row sums of A345272. - _Omar E. Pol_, Jun 14 2021
%C Also the alternating row sums of A353690. - _Omar E. Pol_, Jun 05 2022
%H Alois P. Heinz, <a href="/A175254/b175254.txt">Table of n, a(n) for n = 1..10000</a> (first 2209 terms from Indranil Ghosh)
%F Conjecture: a(n) = Sum_{k=0..n} A006218(n-k). - _R. J. Mathar_, Oct 17 2012
%F a(n) = A000330(n) - A072481(n). - _Omar E. Pol_, Aug 12 2015
%F a(n) ~ Pi^2*n^3/36. - _Vaclav Kotesovec_, Sep 25 2016
%F G.f.: (1/(1 - x)^2)*Sum_{k>=1} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 03 2017
%F a(n) = Sum_{k=1..n} Sum_{i=1..k} k - (k mod i). - _Wesley Ivan Hurt_, Sep 13 2017
%F a(n) = A244050(n)/4. - _Omar E. Pol_, Jan 22 2021
%F a(n) = (n+1)*A024916(n) - A143128(n). - _Vaclav Kotesovec_, May 11 2022
%e For n = 4: a(4) = sigma(1)*4 + sigma(2)*3 + sigma(3)*2 + sigma(4)*1 = 1*4 + 3*3 + 4*2 + 7*1 = 28.
%p b:= proc(n) option remember; `if`(n<1, [0$2],
%p (p-> p+[numtheory[sigma](n), p[1]])(b(n-1)))
%p end:
%p a:= n-> b(n+1)[2]:
%p seq(a(n), n=1..45); # _Alois P. Heinz_, Oct 07 2021
%t Table[Sum[DivisorSigma[1, k] (n - k + 1), {k, n}], {n, 45}] (* _Michael De Vlieger_, Nov 24 2015 *)
%o (PARI) a(n) = sum(x=1, n, sigma(x)*(n-x+1)) \\ _Michel Marcus_, Mar 18 2013
%o (Python)
%o from math import isqrt
%o def A175254(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*(2*s+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+2*q+1)+3*(n+1)*(2*k+q+1)) for k in range(1,s+1)))//6 # _Chai Wah Wu_, Oct 21 2023
%Y Cf. A000203, A000217, A006218, A024916, A072481, A237593, A244050, A245092, A262612, A274824, A319073, A340793, A345272, A353690.
%Y Cf. A143128, A353908.
%K nonn,easy
%O 1,2
%A _Jaroslav Krizek_, Mar 14 2010
%E Corrected by _Jaroslav Krizek_, Mar 17 2010
%E More terms from _Michel Marcus_, Mar 18 2013
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