A143128 a(n) = Sum_{k=1..n} k*sigma(k).

1, 7, 19, 47, 77, 149, 205, 325, 442, 622, 754, 1090, 1272, 1608, 1968, 2464, 2770, 3472, 3852, 4692, 5364, 6156, 6708, 8148, 8923, 10015, 11095, 12663, 13533, 15693, 16685, 18701, 20285, 22121, 23801, 27077, 28483, 30763, 32947, 36547

Offset: 1

Author

Gary W. Adamson, Jul 26 2008

Keywords

• nonn

Comments

Partial sums of A064987. - Omar E. Pol, Jul 04 2014

a(n) is also the volume after n-th step of the symmetric staircase described in A244580 (see also A237593). - Omar E. Pol, Jul 31 2018

In general, for j >= 1 and m >= 0, Sum_{k=1..n} k^m * sigma_j(k) ~ n^(j+m+1) * zeta(j+1) / (j+m+1). - Daniel Suteu, Nov 21 2018

Examples

			a(4) = 47 = (1 + 6 + 12 + 28) where A064987 = (1, 6, 12, 28, 30, ...).
a(4) = 47 = sum of row 4 terms of triangle A110662 = (15 + 14 + 11 + 7).
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..4267

Crossrefs

Cf. A000203, A064987, A110662, A175254, A237593, A244580, A256533.

Programs

Magma

[(&+[k*DivisorSigma(1,k): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Nov 21 2018

Maple

with(numtheory): a:=proc(n) options operator, arrow: sum(k*sigma(k), k=1..n) end proc: seq(a(n),n=1..40); # Emeric Deutsch, Aug 12 2008

Mathematica

Table[Sum[i*DivisorSigma[1, i], {i, n}], {n, 50}] (* Wesley Ivan Hurt, Jul 06 2014 *)

PARI

a(n)=sum(k=1,n,k*sigma(k)) \\ Charles R Greathouse IV, Apr 27 2015

PARI

f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
g(n) = n*(n+1)/2; \\ A000217
a(n) = sum(k=1, sqrtint(n), k * f(n\k) + k^2 * g(n\k)) - f(sqrtint(n)) * g(sqrtint(n)); \\ Daniel Suteu, Nov 26 2020

Python

def A143128(n): return sum(k**2*(m:=n//k)*(m+1)>>1 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

Python

from math import isqrt
def A143128(n): return ((-((s:=isqrt(n))*(s+1))**2*(2*s+1)>>1) + sum((q:=n//k)*(q+1)*k*(3*k+2*q+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

Sage

[sum(k*sigma(k,1) for k in (1..n)) for n in (1..50)] # G. C. Greubel, Nov 21 2018

Formula

Sum {k=1..n} k*sigma(k), where sigma(n) = A000203: (1, 3, 4, 7, 6, 12, ...) and n*sigma(n) = A064987: (1, 6, 12, 28, ...). Equals row sums of triangle A110662. - Emeric Deutsch, Aug 12 2008

a(n) ~ n^3 * Pi^2/18. - Charles R Greathouse IV, Jun 19 2012

G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

a(n) = Sum_{k=1..n} k^2/2 * floor(n/k) * floor(1 + n/k). - Daniel Suteu, May 29 2018

a(n) = A256533(n) - A175254(n-1), n >= 2. - Omar E. Pol, Jul 31 2018

a(n) = Sum_{k=1..s} (k*A000330(floor(n/k)) + k^2*A000217(floor(n/k))) - A000330(s)*A000217(s), where s = floor(sqrt(n)). - Daniel Suteu, Nov 26 2020

a(n) = Sum_{k=1..n} Sum_{i=1..floor(n/k)} i*k^2. - Wesley Ivan Hurt, Nov 26 2020

Extensions

Corrected and extended by Emeric Deutsch, Aug 12 2008