A292962 Rectangular array by antidiagonals: T(n,m) = rank of n*(r-1+m) when all the numbers k*(r+h), where r = log(2), k>=1, h>=0, are jointly ranked.

1, 3, 2, 5, 7, 4, 9, 14, 13, 6, 11, 21, 24, 19, 8, 16, 29, 36, 35, 26, 10, 18, 38, 50, 53, 46, 32, 12, 23, 45, 63, 72, 68, 59, 41, 15, 27, 56, 77, 90, 94, 87, 73, 47, 17, 30, 65, 92, 110, 119, 117, 106, 84, 54, 20, 34, 74, 107, 132, 146, 150, 142, 125, 98

Offset: 1

Author

Clark Kimberling, Oct 05 2017

Keywords

• nonn
• easy
• tabl

Comments

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

Examples

			Northwest corner:
1    3    5    9    11   16   18
2    7    14   21   29   38   45
4    13   24   36   50   63   77
6    19   35   53   72   90   110
8    26   46   68   94   119  146
10   32   59   87   117  150  181
12   41   73   106  142  180  219
The numbers k*(r+h), approximately:
(for k=1):   0.693   1.693   2.693 ...
(for k=2):   1.386   3.386   5.386 ...
(for k=3):   2.079   5.079   8.079 ...
Replacing each by its rank gives
1    3    5
2    7    14
4    13   24
		

Links

• Clark Kimberling, Antidiagonals n=1..60, flattened

Crossrefs

Cf. A182801.

Programs

Mathematica

r = Log[2]; z = 12;
t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292962 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292962 sequence *)

Formula

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=log(2) and [ ]=floor.