A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23

Offset: 1

Author

Clark Kimberling, Dec 14 2010

Keywords

• nonn
• tabl

Comments

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).

The monotonic ordering of this sequence, prefixed by 1, is A000961.

The joint-rank array of this sequence is A182869.

Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

Examples

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
• Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Crossrefs

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).

Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).

A319075 extends the array with 0th powers.

Subtable of A242378, A284457, A329332.

Programs

Mathematica

TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

Formula

From Peter Munn, Dec 29 2019: (Start)

A(i,j) = A182945(j,i) = A319075(j,i).

A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).

A(i,i) = A062457(i).

(End)

Extensions

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022