A306533 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} floor(n/j^k).

1, 1, 4, 1, 3, 9, 1, 2, 5, 16, 1, 2, 3, 8, 25, 1, 2, 3, 5, 10, 36, 1, 2, 3, 4, 6, 14, 49, 1, 2, 3, 4, 5, 7, 16, 64, 1, 2, 3, 4, 5, 6, 8, 20, 81, 1, 2, 3, 4, 5, 6, 7, 10, 23, 100, 1, 2, 3, 4, 5, 6, 7, 9, 12, 27, 121, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 29, 144, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 35, 169

Offset

1,3

Links

Table of n, a(n) for n=1..91.

Formula

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

Example

Square array begins:
   1,   1,  1,  1,  1,  1,  ...
   4,   3,  2,  2,  2,  2,  ...
   9,   5,  3,  3,  3,  3,  ...
  16,   8,  5,  4,  4,  4,  ...
  25,  10,  6,  5,  5,  5,  ...
  36,  14,  7,  6,  6,  6,  ...

Mathematica

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

Crossrefs

Columns k=0..4 give A000290, A006218, A013936, A013937, A013938.

Cf. A306534.

Sequence in context: A226574 A331154 A331150 * A331146 A196770 A154182

Adjacent sequences: A306530 A306531 A306532 * A306534 A306535 A306536

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Feb 22 2019

Status

approved