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A143128 a(n) = Sum_{k=1..n} k*sigma(k). 8
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 (list; graph; refs; listen; history; text; internal format)
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) = A256533(n) - A175254(n-1), n >= 2. - Omar E. Pol, Jul 31 2018

a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (Bernoulli(3, 1+floor(n/k)) - Bernoulli(3, 1+floor(n/(k+1))))/3 + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} k^2 * floor(n/k) * floor(1+n/k))/2, where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 23 2018

EXAMPLE

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).

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) a(n) = (sum(k=1, sqrtint(n), k*(k+1) * ((n\k)^3/3 + (n\k)^2/2 + (n\k)/6 - (n\(k+1))^3/3 - (n\(k+1))^2/2 - (n\(k+1))/6)) + sum(k=1, n\(sqrtint(n)+1), k*k * (n\k) * (1+(n\k))))/2; \\ Daniel Suteu, Nov 23 2018

(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

CROSSREFS

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

Sequence in context: A155273 A277613 A278403 * A238730 A139865 A146403

Adjacent sequences:  A143125 A143126 A143127 * A143129 A143130 A143131

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jul 26 2008

EXTENSIONS

Corrected and extended by Emeric Deutsch, Aug 12 2008

STATUS

approved

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Last modified July 21 02:02 EDT 2019. Contains 325189 sequences. (Running on oeis4.)