OFFSET
1,2
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
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..4267
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) = 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
EXAMPLE
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 *)
PROG
(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
(Magma) [(&+[k*DivisorSigma(1, k): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Nov 21 2018
(Sage) [sum(k*sigma(k, 1) for k in (1..n)) for n in (1..50)] # G. C. Greubel, Nov 21 2018
(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
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jul 26 2008
EXTENSIONS
Corrected and extended by Emeric Deutsch, Aug 12 2008
STATUS
approved