%I #25 May 09 2021 02:51:05

%S 1,1,3,1,3,4,1,3,4,7,1,3,4,9,6,1,3,4,13,6,12,1,3,4,21,6,24,8,1,3,4,37,

%T 6,66,8,15,1,3,4,69,6,216,8,41,13,1,3,4,133,6,762,8,201,37,18,1,3,4,

%U 261,6,2784,8,1289,253,68,12,1,3,4,517,6,10386,8,9225,2197,648,12,28

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

%H Seiichi Manyama, <a href="/A308690/b308690.txt">Antidiagonals n = 1..140, flattened</a>

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

%F A(p,k) = p+1 for prime p.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 3, 3, 3, 3, 3, 3, 3, ...

%e 4, 4, 4, 4, 4, 4, 4, ...

%e 7, 9, 13, 21, 37, 69, 133, ...

%e 6, 6, 6, 6, 6, 6, 6, ...

%e 12, 24, 66, 216, 762, 2784, 10386, ...

%e 8, 8, 8, 8, 8, 8, 8, ...

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

%Y Columns k=0..3 give A000203, A055225, A308688, A308689.

%Y Cf. A294579, A308509, A308694.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Jun 17 2019