%I #13 Oct 23 2023 15:19:49
%S 8,56,189,491,1007,1930,3276,5314,8082,11973,16783,23355,31314,41380,
%T 53566,68510,85771,106981,130973,159470,192020,229762,271873,320779,
%U 375031,436311,504464,581422,664364,759025,860907,973989,1097783
%N a(n) = Sum of all divisors of all numbers < (n+1)^2.
%C Partial sums of A168012.
%H Chai Wah Wu, <a href="/A168013/b168013.txt">Table of n, a(n) for n = 1..10000</a>
%e For n=2 the a(2)=56 because the numbers < (2+1)^2 are 1,2,3,4,5,6,7 and 8. Then a(2)= sigma(1))+sigma(2)+sigma(3)+sigma(4)+sigma(5)+sigma(6)+sigma(7)+sigma(8) = 1+3+4+7+6+12+8+15 = 56, where sigma(n) is the sum of divisor of n (see A000203).
%t A168012[n_]:=Sum[DivisorSigma[1,k],{k,n^2,(n+1)^2-1}];
%t Accumulate[Array[A168012,50]] (* _Paolo Xausa_, Oct 23 2023 *)
%o (Python)
%o def A168013(n):
%o m = n*(n+2)
%o return sum((q:=m//k)*((k<<1)+q+1) for k in range(1,n+1))-n**2*(n+1)>>1 # _Chai Wah Wu_, Oct 23 2023
%Y Cf. A000203, A024916, A168010, A168011, A168012.
%K nonn
%O 1,1
%A _Omar E. Pol_, Nov 16 2009
%E More terms from _Sean A. Irvine_, Dec 07 2009
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