A319086 -id:A319086 - cp's OEIS frontend

A356125 a(n) = Sum_{k=1..n} k * sigma_2(k).

1, 11, 41, 125, 255, 555, 905, 1585, 2404, 3704, 5046, 7566, 9776, 13276, 17176, 22632, 27562, 35752, 42630, 53550, 64050, 77470, 89660, 110060, 126335, 148435, 170575, 199975, 224393, 263393, 293215, 336895, 377155, 426455, 471955, 540751, 591441, 660221, 726521

Offset: 1

Seiichi Manyama, Jul 27 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Partial sums of A328259.
Column k=3 of A356124.
Cf. A319086, A356042.

a[n_] := Sum[k * DivisorSigma[2, k], {k, 1, n}]; Array[a, 39] (* Amiram Eldar, Jul 28 2022 *)

PARI
```
a(n) = sum(k=1, n, k*sigma(k, 2));
```

a(n) = sum(k=1, n, k^3*binomial(n\k+1, 2));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-x^k)^2)/(1-x))

from math import isqrt
def A356125(n): return (-((s:=isqrt(n))*(s+1))**3>>1) + sum(k*(q:=n//k)*(q+1)*(2*k**2+q*(q+1)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^3 * binomial(floor(n/k)+1,2).
G.f.: (1/(1-x)) * Sum_{k>=1} k^3 * x^k/(1 - x^k)^2.
a(n) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Aug 02 2022

A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

Original entry on oeis.org

1, 16, 62, 219, 405, 1053, 1523, 2948, 4407, 7041, 8703, 15283, 17949, 24657, 32685, 44806, 50536, 70687, 78573, 105411, 125879, 149879, 163565, 222425, 247476, 286134, 327634, 396258, 423084, 532236, 564818, 664763, 738095, 821693, 904937, 1107618, 1162268, 1277588, 1395760

Offset: 1

Seiichi Manyama, Jul 31 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=3 of A356250.
Cf. A000537, A024916, A318742, A350123.
Cf. A064603, A356125, A319086.

a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jul 31 2022 *)

PARI
```
a(n) = sum(k=1, n, (k*(n\k))^3);
```

a(n) = sum(k=1, n, k^3*sumdiv(k, d, 1-(1-1/d)^3));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))

from math import isqrt
def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^3 * Sum_{d|k} (1 - (1 - 1/d)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
From Vaclav Kotesovec, Aug 02 2022: (Start)
a(n) = A064603(n) - 3*A356125(n) + 3*A319086(n).
a(n) ~ n^4 * (Pi^2/8 + Pi^4/360 - 3*zeta(3)/4). (End)

A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

Original entry on oeis.org

1, 25, 133, 581, 1331, 3923, 6667, 14347, 23824, 41824, 57796, 106180, 136938, 202794, 283794, 410770, 499204, 726652, 863832, 1199832, 1496184, 1879512, 2171520, 3000960, 3485335, 4223527, 5010847, 6240159, 6971829, 8915829, 9869141, 11933525, 13658501

Offset: 1

Seiichi Manyama, Oct 20 2023

Wikipedia, Faulhaber's formula.

Partial sums of A282211.
Cf. A319086, A320895, A356125, A364269.
Cf. A000203, A000537, A143128.

Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)

f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=1) = sum(k=1, n, k^(s+t)*f(n\k, s));

def A364194(n): return sum((k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

from math import isqrt
def A364194(n): return ((((s:=isqrt(n))*(s + 1))**3*(2*s+1)*(1-3*s*(s+1))>>1) + sum((q:=n//k)*(q+1)*k**3*(q*(15*k+q*(15*k+12*q+18)+2)-2) for k in range(1,s+1)))//60 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^4 * A000537(floor(n/k)).

a(n) ~ (zeta(2)/5) * n^5. - Amiram Eldar, Oct 20 2023

cp's OEIS Frontend

A356125 a(n) = Sum_{k=1..n} k * sigma_2(k).

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A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

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Author

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Mathematica

PARI

PARI

PARI

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Formula

A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula