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A256533 Product of n and the sum of all divisors of all positive integers <= n.

%I A256533 #63 Oct 22 2023 16:36:30
%S A256533 1,8,24,60,105,198,287,448,621,870,1089,1524,1833,2310,2835,3520,4046,
%T A256533 4986,5643,6780,7791,8954,9913,11784,13050,14664,16308,18480,20010,
%U A256533 22860,24614,27424,29865,32606,35245,39528,42032,45448,48828,53680,56744,62160,65532,70752,75870,80868,84882,92640,97363,104000
%N A256533 Product of n and the sum of all divisors of all positive integers <= n.
%C A256533 a(n) is also sum of the volumes (or the total number of unit cubes) from two complementary polycubes: the irregular staircase after n-th stage described in A244580, and the irregular stepped pyramid after (n-1)st stage described in A245092. Note that in both structures the horizontal area in the n-th level is also the symmetric representation of sigma(n). This comment is represented by the third formula.
%H A256533 Indranil Ghosh, <a href="/A256533/b256533.txt">Table of n, a(n) for n = 1..10000</a>
%F A256533 a(n) = n*A024916(n).
%F A256533 a(n) = n^3 - A256532(n).
%F A256533 a(n) = A143128(n) + A175254(n-1), n > 1.
%F A256533 a(n) = A332264(n) + A175254(n). - _Omar E. Pol_, Apr 29 2020
%e A256533 For n = 3; a(3) = 3 * 8 = 19 + 5 = 24.
%t A256533 a[n_]:=n*Apply[Plus,Flatten[Divisors[Range[n]]]]; Array[a,50] (* _Ivan N. Ianakiev_, May 03 2015 *)
%t A256533 nxt[{n_,sd_,a_}]:=Module[{k=(n+1)*(DivisorSigma[1,n+1]+sd)},{n+1,sd+DivisorSigma[ 1,n+1],k}]; NestList[ nxt,{1,1,1},50][[;;,3]] (* _Harvey P. Dale_, Jun 12 2023 *)
%o A256533 (Python)
%o A256533 def A256533(n):
%o A256533     s=0
%o A256533     for k in range(1, n+1):
%o A256533         s+=n%k
%o A256533     return (n**3)-(s*n) # _Indranil Ghosh_, Feb 13 2017
%o A256533 (Python)
%o A256533 from math import isqrt
%o A256533 def A256533(n): return n*(-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # _Chai Wah Wu_, Oct 22 2023
%o A256533 (PARI) a(n) = n*sum(k=1, n, n\k*k); \\ _Michel Marcus_, Apr 29 2020
%Y A256533 Cf. A000203, A000578, A024916, A064987, A143128, A175254, A196020, A236104, A237270, A237271, A237593, A244580, A245092, A245100, A256532, A332264.
%K A256533 nonn
%O A256533 1,2
%A A256533 _Omar E. Pol_, May 02 2015