OFFSET
1,5
COMMENTS
LINKS
Seiichi Manyama, Antidiagonals n = 1..140, flattened
FORMULA
A(n,k) = Sum_{d|n} k^omega(d). - Seiichi Manyama, May 15 2021
EXAMPLE
Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
n=1: 1 1 1 1 1 1 1 1
n=2: 1 2 3 4 5 6 7 8
n=3: 1 2 3 4 5 6 7 8
n=4: 1 3 5 7 9 11 13 15
n=5: 1 2 3 4 5 6 7 8
n=6: 1 4 9 16 25 36 49 64
n=7: 1 2 3 4 5 6 7 8
n=8: 1 4 7 10 13 16 19 22
n=9: 1 3 5 7 9 11 13 15
Triangle begins:
1
1 1
1 2 1
1 3 2 1
1 4 3 3 1
1 5 4 5 2 1
1 6 5 7 3 4 1
1 7 6 9 4 9 2 1
1 8 7 11 5 16 3 4 1
1 9 8 13 6 25 4 7 3 1
1 10 9 15 7 36 5 10 5 4 1
1 11 10 17 8 49 6 13 7 9 2 1
1 12 11 19 9 64 7 16 9 16 3 6 1
1 13 12 21 10 81 8 19 11 25 4 15 2 1
For example, row n = 8 counts the following divisors:
1 64 243 256 125 36 7 1
32 81 128 25 18 1
16 27 64 5 12
8 9 32 1 9
4 3 16 6
2 1 8 4
1 4 3
2 2
1 1
MATHEMATICA
Table[DivisorSigma[0, k^(n-k)], {n, 10}, {k, n}]
PROG
(PARI) A(n, k) = numdiv(n^k); \\ Seiichi Manyama, May 15 2021
CROSSREFS
Columns k=1..9 of the array give A000005, A048691, A048785, A344327, A344328, A344329, A343526, A344335, A344336.
Row n = 6 of the array is A000290.
Diagonal n = k of the array is A062319.
Array antidiagonal sums (row sums of the triangle) are A343657.
Dominated by A343658.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 28 2021
STATUS
approved