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

0, 1, 5, 12, 26, 44, 73, 109, 157, 215, 292, 375, 481, 603, 744, 900, 1087, 1287, 1522, 1773, 2053, 2361, 2712, 3073, 3476, 3913, 4389, 4891, 5448, 6021, 6653, 7316, 8028, 8786, 9599, 10427, 11326, 12277, 13287, 14325, 15442, 16587, 17815, 19089, 20418, 21811

Offset: 1

Author

Darío Clavijo, Nov 15 2023

Keywords

• nonn

Comments

Partial sums of A049766.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000292, A002411, A049766, A072481, A175254.

Programs

Maple

N:= 100: # for a(1)..a(N)
S:= [seq(NumberTheory:-SumOfDivisors(i,1),i=1..N+1)]:
SS:= ListTools:-PartialSums(S):
S2:= [seq(i*S[i],i=1..N+1)]:
SS2:= ListTools:-PartialSums(S2):
f:= n -> 1/2*n^2*(n+1) - (n+1)*SS[n+1]+SS2[n+1]:
map(f, [$1..N]); # Robert Israel, Dec 20 2023

Mathematica

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 *)

PARI

a(n) = sum(j=1, n, sum(i=1, n, j % i)); \\ Michel Marcus, Nov 16 2023

Python

from sympy import divisor_sigma
A002411 = lambda n: ((n*n)*(n+1))>>1
A175254 = lambda n: sum(divisor_sigma(i) * (n-i+1) for i in range(1,n+1))
a = lambda n: A002411(n) - A175254(n)
print([a(n) for n in range(1, 53)])

Python

from math import isqrt
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

Formula

a(n) = A072481(n) + A000292(n-1).

a(n) = A002411(n) - A175254(n).