A308694 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*(n/d - 1)), read by antidiagonals.

1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 1, 2, 2, 6, 2, 4, 1, 2, 2, 10, 2, 9, 2, 1, 2, 2, 18, 2, 27, 2, 4, 1, 2, 2, 34, 2, 93, 2, 14, 3, 1, 2, 2, 66, 2, 339, 2, 82, 11, 4, 1, 2, 2, 130, 2, 1269, 2, 578, 83, 23, 2, 1, 2, 2, 258, 2, 4827, 2, 4354, 731, 283, 2, 6

Offset

1,3

Links

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

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j^(k+1))).

A(p,k) = 2 for prime p.

Example

Square array begins:
   1, 1,  1,  1,   1,    1,    1, ...
   2, 2,  2,  2,   2,    2,    2, ...
   2, 2,  2,  2,   2,    2,    2, ...
   3, 4,  6, 10,  18,   34,   66, ...
   2, 2,  2,  2,   2,    2,    2, ...
   4, 9, 27, 93, 339, 1269, 4827, ...
   2, 2,  2,  2,   2,    2,    2, ...

Mathematica

T[n_, k_] := DivisorSum[n, #^(k*(n/# - 1)) &]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

Crossrefs

Columns k=0..3 give A000005, A087909, A308692, A308693.

Cf. A308509, A308690.

Sequence in context: A053260 A267135 A140223 * A280521 A278043 A014643

Adjacent sequences: A308691 A308692 A308693 * A308695 A308696 A308697

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 17 2019

Status

approved