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A367379 a(n) = Sum_{j=1..n} Sum_{i=1..n} (j mod i).

%I A367379 #34 Dec 20 2023 14:55:22
%S A367379 0,1,5,12,26,44,73,109,157,215,292,375,481,603,744,900,1087,1287,1522,
%T A367379 1773,2053,2361,2712,3073,3476,3913,4389,4891,5448,6021,6653,7316,
%U A367379 8028,8786,9599,10427,11326,12277,13287,14325,15442,16587,17815,19089,20418,21811
%N A367379 a(n) = Sum_{j=1..n} Sum_{i=1..n} (j mod i).
%C A367379 Partial sums of A049766.
%H A367379 Robert Israel, <a href="/A367379/b367379.txt">Table of n, a(n) for n = 1..10000</a>
%F A367379 a(n) = A072481(n) + A000292(n-1).
%F A367379 a(n) = A002411(n) - A175254(n).
%p A367379 N:= 100: # for a(1)..a(N)
%p A367379 S:= [seq(NumberTheory:-SumOfDivisors(i,1),i=1..N+1)]:
%p A367379 SS:= ListTools:-PartialSums(S):
%p A367379 S2:= [seq(i*S[i],i=1..N+1)]:
%p A367379 SS2:= ListTools:-PartialSums(S2):
%p A367379 f:= n -> 1/2*n^2*(n+1) - (n+1)*SS[n+1]+SS2[n+1]:
%p A367379 map(f, [$1..N]); # _Robert Israel_, Dec 20 2023
%t A367379 a[n_]:=n^2(n+1)/2-Sum[DivisorSigma[1,i](n-i+1),{i,n+1}]; Array[a,47,0] (* _Stefano Spezia_, Nov 17 2023 *)
%o A367379 (Python)
%o A367379 from sympy import divisor_sigma
%o A367379 A002411 = lambda n: ((n*n)*(n+1))>>1
%o A367379 A175254 = lambda n: sum(divisor_sigma(i) * (n-i+1) for i in range(1,n+1))
%o A367379 a = lambda n: A002411(n) - A175254(n)
%o A367379 print([a(n) for n in range(1, 53)])
%o A367379 (Python)
%o A367379 from math import isqrt
%o A367379 def A367379(n): return (n**2*(n+1)>>1)-(((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_, Dec 20 2023
%o A367379 (PARI) a(n) = sum(j=1, n, sum(i=1, n, j % i)); \\ _Michel Marcus_, Nov 16 2023
%Y A367379 Cf. A000292, A002411, A049766, A072481, A175254.
%K A367379 nonn
%O A367379 1,3
%A A367379 _Darío Clavijo_, Nov 15 2023