A175254 a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.

1, 5, 13, 28, 49, 82, 123, 179, 248, 335, 434, 561, 702, 867, 1056, 1276, 1514, 1791, 2088, 2427, 2798, 3205, 3636, 4127, 4649, 5213, 5817, 6477, 7167, 7929, 8723, 9580, 10485, 11444, 12451, 13549, 14685, 15881, 17133, 18475, 19859, 21339, 22863, 24471, 26157

Offset: 1

Author

Jaroslav Krizek, Mar 14 2010

Keywords

• nonn
• easy

Comments

Partial sums of A024916. - Omar E. Pol, Jul 03 2014

a(n) is also the volume of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Aug 12 2015

Also the alternating row sums of A262612. - Omar E. Pol, Nov 23 2015

From Omar E. Pol, Jan 20 2021: (Start)

Convolution of A000203 and A000027.

Convolution of A340793 and the nonzero terms of A000217.

Antidiagonal sums of A319073.

Row sums of A274824. (End)

Row sums of A345272. - Omar E. Pol, Jun 14 2021

Also the alternating row sums of A353690. - Omar E. Pol, Jun 05 2022

Examples

			For n = 4: a(4) = sigma(1)*4 + sigma(2)*3 + sigma(3)*2 + sigma(4)*1 = 1*4 + 3*3 + 4*2 + 7*1 = 28.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2209 terms from Indranil Ghosh)

Crossrefs

Cf. A000203, A000217, A006218, A024916, A072481, A237593, A244050, A245092, A262612, A274824, A319073, A340793, A345272, A353690.

Cf. A143128, A353908.

Programs

Maple

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[sigma](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Oct 07 2021

Mathematica

Table[Sum[DivisorSigma[1, k] (n - k + 1), {k, n}], {n, 45}] (* Michael De Vlieger, Nov 24 2015 *)

PARI

a(n) = sum(x=1, n, sigma(x)*(n-x+1)) \\ Michel Marcus, Mar 18 2013

Python

from math import isqrt
def A175254(n): return (((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, Oct 21 2023

Formula

Conjecture: a(n) = Sum_{k=0..n} A006218(n-k). - R. J. Mathar, Oct 17 2012

a(n) = A000330(n) - A072481(n). - Omar E. Pol, Aug 12 2015

a(n) ~ Pi^2*n^3/36. - Vaclav Kotesovec, Sep 25 2016

G.f.: (1/(1 - x)^2)*Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017

a(n) = Sum_{k=1..n} Sum_{i=1..k} k - (k mod i). - Wesley Ivan Hurt, Sep 13 2017

a(n) = A244050(n)/4. - Omar E. Pol, Jan 22 2021

a(n) = (n+1)*A024916(n) - A143128(n). - Vaclav Kotesovec, May 11 2022

Extensions

Corrected by Jaroslav Krizek, Mar 17 2010

More terms from Michel Marcus, Mar 18 2013