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

%I A182944 #52 Jul 18 2023 07:37:57
%S A182944 2,4,3,8,9,5,16,27,25,7,32,81,125,49,11,64,243,625,343,121,13,128,729,
%T A182944 3125,2401,1331,169,17,256,2187,15625,16807,14641,2197,289,19,512,
%U A182944 6561,78125,117649,161051,28561,4913,361,23
%N A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.
%C A182944 We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
%C A182944 The monotonic ordering of this sequence, prefixed by 1, is A000961.
%C A182944 The joint-rank array of this sequence is A182869.
%C A182944 Main diagonal gives A062457. - _Omar E. Pol_, Sep 11 2018
%H A182944 Michael De Vlieger, <a href="/A182944/b182944.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150, flattened)
%H A182944 Michael De Vlieger, <a href="/A182944/a182944.png">Diagram</a> 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.
%F A182944 From _Peter Munn_, Dec 29 2019: (Start)
%F A182944 A(i,j) = A182945(j,i) = A319075(j,i).
%F A182944 A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
%F A182944 A(i,i) = A062457(i).
%F A182944 (End)
%e A182944 Square array A(i,j) begins:
%e A182944   i \ j: 1      2      3      4      5  ...
%e A182944   ---\-------------------------------------
%e A182944   1:     2,     4,     8,    16,    32, ...
%e A182944   2:     3,     9,    27,    81,   243, ...
%e A182944   3:     5,    25,   125,   625,  3125, ...
%e A182944   4:     7,    49,   343,  2401, 16807, ...
%e A182944   ...
%e A182944 The triangle T(n,k) begins:
%e A182944   n\k:  1     2     3     4     5     6  ...
%e A182944   1:    2
%e A182944   2:    4     3
%e A182944   3:    8     9     5
%e A182944   4:   16    27    25     7
%e A182944   5:   32    81   125    49    11
%e A182944   6:   64   243   625   343   121    13
%e A182944   ...
%t A182944 TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]
%Y A182944 Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
%Y A182944 Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
%Y A182944 A319075 extends the array with 0th powers.
%Y A182944 Subtable of A242378, A284457, A329332.
%K A182944 nonn,tabl
%O A182944 1,1
%A A182944 _Clark Kimberling_, Dec 14 2010
%E A182944 Clarified in respect of alternate reading as a triangle by _Peter Munn_, Aug 28 2022