A308704 -id:A308704 - cp's OEIS frontend

A308698 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 261, 2, 1, 257, 19684, 65553, 3126, 4, 1, 1025, 531442, 16777281, 9765626, 46688, 2, 1, 4097, 14348908, 4294967553, 30517578126, 2176783082, 823544, 4, 1, 16385, 387420490, 1099511628801, 95367431640626, 101559956688164, 678223072850, 16777477, 3

Offset: 1

Seiichi Manyama, Jun 17 2019

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  261,   65553,    16777281,     4294967553, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened

Columns k=0..3 give A000005, A062796, A308696, A308697.
Row n=1..2 give A000012, A052539.
Cf. A308509, A308701, A308704.

T[n_, k_] := DivisorSum[n, #^(k*#) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

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

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

A308701 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*(d-1)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 10, 3, 1, 9, 82, 67, 2, 1, 17, 730, 4101, 626, 4, 1, 33, 6562, 262153, 390626, 7788, 2, 1, 65, 59050, 16777233, 244140626, 60466262, 117650, 4, 1, 129, 531442, 1073741857, 152587890626, 470184985314, 13841287202, 2097219, 3

Offset: 1

Seiichi Manyama, Jun 22 2019

			Square array begins:
   1,   1,      1,         1,            1, ...
   2,   3,      5,         9,           17, ...
   2,  10,     82,       730,         6562, ...
   3,  67,   4101,    262153,     16777233, ...
   2, 626, 390626, 244140626, 152587890626, ...

Seiichi Manyama, Antidiagonals n = 1..53, flattened

Columns k=0..2 give A000005, A262843, A308753.
Row n=1..3 give A000012, A000051, A062396.
Cf. A308694, A308698, A308704.

T[n_, k_] := DivisorSum[n, #^(k*(# - 1)) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

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

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

cp's OEIS Frontend

A308698 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).

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