A350108 -id:A350108 - cp's OEIS frontend

A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

1, 1, 4, 1, 6, 8, 1, 10, 14, 15, 1, 18, 32, 31, 21, 1, 34, 86, 87, 45, 33, 1, 66, 248, 295, 153, 81, 41, 1, 130, 734, 1095, 669, 309, 101, 56, 1, 258, 2192, 4231, 3201, 1521, 443, 150, 69, 1, 514, 6566, 16647, 15765, 8373, 2633, 722, 191, 87, 1, 1026, 19688, 66055, 78393, 48321, 17411, 4746, 1005, 253, 99

Offset: 1

Seiichi Manyama, Dec 14 2021

			Square array begins:
   1,   1,   1,    1,     1,      1,      1, ...
   4,   6,  10,   18,    34,     66,    130, ...
   8,  14,  32,   86,   248,    734,   2192, ...
  15,  31,  87,  295,  1095,   4231,  16647, ...
  21,  45, 153,  669,  3201,  15765,  78393, ...
  33,  81, 309, 1521,  8373,  48321, 284709, ...
  41, 101, 443, 2633, 17411, 119321, 828323, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..3 give A024916, A350107, A350108.
T(n,n) gives A350109.
Cf. A319649, A344725.

T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 14 2021 *)

PARI
```
T(n, k) = sum(j=1, n, j*(n\j)^k);
```

T(n, k) = sum(j=1, n, j*sumdiv(j, d, (d^k-(d-1)^k)/d));

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j)^2.

T(n,k) = Sum_{j=1..n} j * Sum_{d|j} (d^k - (d - 1)^k)/d.

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

Original entry on oeis.org

1, 12, 40, 121, 207, 473, 649, 1142, 1611, 2401, 2853, 4647, 5285, 6879, 8759, 11452, 12558, 16739, 18127, 23353, 27129, 31171, 33219, 43573, 47524, 53210, 59538, 69996, 73274, 89694, 93446, 107195, 116731, 126545, 137505, 164580, 169946, 182244, 195644, 225454

Offset: 1

Seiichi Manyama, Dec 15 2021

nonn

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

Cf. A318742, A350108, A350123, A350125.

Cf. A064602, A143128, A319085.

Accumulate[Table[DivisorSigma[2, k] - 3*k*DivisorSigma[1, k] + 3*k^2*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 17 2021 *)

PARI
```
a(n) = sum(k=1, n, k^2*(n\k)^3);
```

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^3-(d-1)^3)/d^2));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))

from math import isqrt
def A350124(n): return (-(s:=isqrt(n))**4*(s+1)*(2*s+1) + sum((q:=n//k)*(k*(3*(k-1))+q*(k*(9*(k-1))+q*(k*(12*k-6)+2)+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^3 - (d - 1)^3)/d^2.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + x^k)/(1 - x^k)^3.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A064602(n) - 3*A143128(n) + 3*A319085(n).
a(n) ~ n^3 * (log(n) + 2*gamma + (zeta(3) - 1)/3 - Pi^2/6), where gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

Formula