A308690 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.
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, 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, 261, 6, 2784, 8, 1289, 253, 68, 12, 1, 3, 4, 517, 6, 10386, 8, 9225, 2197, 648, 12, 28
Offset: 1
Examples
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, 3, ...
4, 4, 4, 4, 4, 4, 4, ...
7, 9, 13, 21, 37, 69, 133, ...
6, 6, 6, 6, 6, 6, 6, ...
12, 24, 66, 216, 762, 2784, 10386, ...
8, 8, 8, 8, 8, 8, 8, ...
Links
- Seiichi Manyama, Antidiagonals n = 1..140, flattened
Programs
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Mathematica
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 *)
Formula
L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j^k)).
A(p,k) = p+1 for prime p.