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

1,3

Comments

Partial sums of A049766.

Links

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

Formula

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A309323 A364970 A294017 * A212561 A199771 A200252

Adjacent sequences: A367376 A367377 A367378 * A367380 A367381 A367382

Keyword

nonn

Author

Darío Clavijo, Nov 15 2023

Status

approved