A279394 Triangle read by rows, T(n,m) = sigma_{n-m}(m) for n >= 1, m = 1,2, ..., n.

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 7, 2, 1, 17, 28, 21, 6, 4, 1, 33, 82, 73, 26, 12, 2, 1, 65, 244, 273, 126, 50, 8, 4, 1, 129, 730, 1057, 626, 252, 50, 15, 3, 1, 257, 2188, 4161, 3126, 1394, 344, 85, 13, 4, 1, 513, 6562, 16513, 15626, 8052, 2402, 585, 91, 18, 2, 1, 1025, 19684, 65793, 78126, 47450, 16808, 4369, 757, 130, 12, 6

Offset: 1

Author

Wolfdieter Lang, Jan 07 2017

Keywords

• nonn
• tabl
• easy

Comments

See A109974 (downward antidiagonals) for details and references. sigma_k(n) is the sum of the k-th power of the positive divisors of n.

This is the triangle read by rows obtained from the array sigma_k(n) for k >= 0, n >= 1, read by upward antidiagonals.

The row sums are A108639.

Examples

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   2
3:  1   3    2
4:  1   5    4    3
5:  1   9   10    7    2
6:  1  17   28   21    6    4
7:  1  33   82   73   26   12   2
8:  1  65  244  273  126   50   8  4
9:  1 129  730 1057  626  252  50 15  3
10: 1 257 2188 4161 3126 1394 344 85 13  4
...
n = 11: 1 513 6562 16513 15626 8052 2402 585 91 18 2,
n = 12: 1 1025 19684 65793 78126 47450 16808 4369 757 130 12 6.
...
		

Crossrefs

Cf. A109974, A108639.

Programs

Maple

T := (n, k) -> numtheory:-sigma[n-k](k):
seq(seq(T(n,k), k=1..n), n=1..12); # Peter Luschny, Jan 07 2017

Mathematica

Table[DivisorSigma[k, #] &[n - k + 1], {n, 0, 11}, {k, n, 0, -1}] (* Michael De Vlieger, Jan 09 2017 *)

Formula

T(n, m) = sigma_{n-m}(m), n >= 1, m = 1..n.