A143127 - cp's OEIS frontend

1, 5, 11, 23, 33, 57, 71, 103, 130, 170, 192, 264, 290, 346, 406, 486, 520, 628, 666, 786, 870, 958, 1004, 1196, 1271, 1375, 1483, 1651, 1709, 1949, 2011, 2203, 2335, 2471, 2611, 2935, 3009, 3161, 3317, 3637, 3719, 4055, 4141, 4405, 4675, 4859, 4953, 5433

Offset: 1

Gary W. Adamson, Jul 26 2008

a(n) is also the sum of all parts of all partitions of all positive integers <= n into equal parts. - Omar E. Pol, May 29 2017

a(n) is also the sum of the multiples of k, not exceeding n, for k = 1, 2, ..., n. See a formula and an example below. - Wolfdieter Lang, Oct 18 2021

			a(3) = 11 = (1 + 4 + 6), where n*d(n) = (1, 4, 6, 12, 10, 24, ...).
a(4) = 23 = (8 + 7 + 5 + 3), where (8, 7, 5, 3) = row 4 of triangle A110661.
a(4) = 23 is the sum of [1 2 3 4|2 4|3|4] (multiples of k=1..4, not exceeding n). - _Wolfdieter Lang_, Oct 18 2021
a(4) = [1] + [2 + 1 + 1] + [3 + 1 + 1 + 1] + [4 + 2 + 2 + 1 + 1 + 1 + 1] = 23. - _Omar E. Pol_, Oct 18 2021

Enrique Pérez Herrero, Table of n, a(n) for n = 1..1000
Jacob F. F. Bulmer, Javier Martínez-Cifuentes, Bryn A. Bell, and Nicolás Quesada, Simulating lossy and partially distinguishable quantum optical circuits: theory, algorithms and applications to experiment validation and state preparation, arXiv:2412.17742 [quant-ph], 2024. See p. 29.
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Partial sums of A038040.
Cf. A000005, A083356.
Row sums of triangle A110661.
Row sums of triangle A143310. - Gary W. Adamson, Aug 06 2008
Cf. A018804.

a143127 n = a143127_list !! (n-1)
a143127_list = scanl1 (+) a038040_list
-- Reinhard Zumkeller, Jan 21 2014

Accumulate[DivisorSigma[0, Range[48]] Range[48]] (* Giovanni Resta, May 29 2018 *)

a(n) = sum(k=1, n, k*numdiv(k)); \\ Michel Marcus, May 29 2018

from math import isqrt
def A143127(n): return -((k:=isqrt(n))*(k+1)>>1)**2+sum(i*(m:=n//i)*(1+m) for i in range(1,k+1)) # Chai Wah Wu, Jul 11 2023

a(n) = Sum_{k=1..n} A038040(k).
a(n) = Sum_{m=1..floor(sqrt(n))} m*(m+floor(n/m))*(floor(n/m)+1-m) - A000330(floor(sqrt(n))) = 2*A083356(n) - A000330(floor(sqrt(n))). - Max Alekseyev, Jan 31 2012
G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017 [Sum_{k>=1} k*x^k/((1-x)*(1-x^k)^2), see A038040. - Wolfdieter Lang, Oct 18 2021]
a(n) = Sum_{k=1..n} k/2 * floor(n/k) * floor(1 + n/k). - Daniel Suteu, May 28 2018
a(n) ~ log(n) * n^2 / 2 + (gamma - 1/4)*n^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 08 2018
From Daniel Hoying, May 21 2020: (Start)
a(n) = (Sum_{i=1..floor(sqrt(n))} i*floor(n/i)*(1+floor(n/i))) - (floor(sqrt(n))*(1+floor(sqrt(n)))/2)^2;
     = (Sum_{i=1..floor(sqrt(n))} i*floor(n/i)*(1+floor(n/i))) - A000537(floor(sqrt(n))).
a(n) = A000537(floor(sqrt(n))) ; n=1;
     = A000537(floor(sqrt(n))) + n*(n+1) - floor(n/2)*(floor(n/2)+1) ; 1     = A000537(floor(sqrt(n))) + n*(n+1) - floor(n/2)*(floor(n/2)+1) + Sum_{i=floor(sqrt(n))+1..floor(n/2)} i*floor(n/i)*(1+floor(n/i)) ; n>=6. (End)
a(n) = Sum_{i=1..n} A018804(i)*floor(n/i). - Ridouane Oudra, Mar 15 2021
a(n) = Sum_{k=1..n} b(n,k), with b(n, k) = Sum_{j=1..floor(n/k)} j*k = k * floor(n/k) * (floor(n/k) + 1)/2. See the formula by Daniel Suteu above. - Wolfdieter Lang, Oct 18 2021

More terms from Carl Najafi, Dec 24 2011

Edited by Max Alekseyev, Jan 31 2012

cp's OEIS Frontend

A143127 a(n) = Sum_{k=1..n} k*d(k) where d(k) is the number of divisors of k.

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