A306533 -id:A306533 - cp's OEIS frontend

A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 10, 1, 33, 83, 74, 32, 14, 1, 65, 245, 274, 136, 52, 16, 1, 129, 731, 1058, 644, 254, 66, 20, 1, 257, 2189, 4162, 3160, 1396, 382, 92, 23, 1, 513, 6563, 16514, 15692, 8054, 2502, 596, 115, 27, 1, 1025, 19685, 65794, 78256, 47452, 17086, 4388, 833, 147, 29

Offset: 1

Seiichi Manyama, May 27 2021

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   5, 11,  29,   83,  245,   731, ...
   8, 22,  74,  274, 1058,  4162, ...
  10, 32, 136,  644, 3160, 15692, ...
  14, 52, 254, 1396, 8054, 47452, ...

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

Columns k=1..5 give A006218, A222548, A318742, A318743, A318744.
T(n,n) gives A332469.
Cf. A306533, A344726.

T[n_, k_] := Sum[Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)

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

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

from math import isqrt
from itertools import count, islice
def A344725_T(n,k): return -(s:=isqrt(n))**(k+1)+sum((q:=n//w)*(w**k-(w-1)**k+q**(k-1)) for w in range(1,s+1))
def A344725_gen(): # generator of terms
     return (A344725_T(k+1,n-k) for n in count(1) for k in range(n))
A344725_list = list(islice(A344725_gen(),30)) # Chai Wah Wu, Oct 26 2023

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

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

A306534 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Sum_{j=0..n} floor(n/k^j).

Original entry on oeis.org

0, 0, 2, 0, 1, 6, 0, 1, 3, 12, 0, 1, 2, 4, 20, 0, 1, 2, 4, 7, 30, 0, 1, 2, 3, 5, 8, 42, 0, 1, 2, 3, 5, 6, 10, 56, 0, 1, 2, 3, 4, 6, 8, 11, 72, 0, 1, 2, 3, 4, 6, 7, 9, 15, 90, 0, 1, 2, 3, 4, 5, 7, 8, 10, 16, 110, 0, 1, 2, 3, 4, 5, 7, 8, 10, 13, 18, 132, 0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 19, 156

Offset: 0

Ilya Gutkovskiy, Feb 22 2019

			Square array begins:
   0,  0,  0,  0,  0,  0,  ...
   2,  1,  1,  1,  1,  1,  ...
   6,  3,  2,  2,  2,  2,  ...
  12,  4,  4,  3,  3,  3,  ...
  20,  7,  5,  5,  4,  4,  ...
  30,  8,  6,  6,  6,  5,  ...

Columns k=1..4 give A002378, A005187, A004128, A087069.

Cf. A306533.

Table[Function[k, Sum[Floor[n/k^j], {j, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

G.f. of column k (for k > 1): (1/(1 - x)) * Sum_{j>=0} x^(k^j)/(1 - x^(k^j)).

cp's OEIS Frontend

A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

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