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

1,1

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

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.

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)

Example

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
  ...

Mathematica

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

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.

Sequence in context: A269851 A284457 A365656 * A269385 A252755 A366275

Adjacent sequences: A182941 A182942 A182943 * A182945 A182946 A182947

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 14 2010

Extensions

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

Status

approved