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%I A365656 #32 Jan 23 2024 16:18:49
%S A365656 1,2,4,3,8,9,5,16,27,25,6,32,81,125,12,7,64,243,625,18,49,10,128,729,
%T A365656 3125,24,343,20,11,256,2187,15625,36,2401,40,121,13,512,6561,78125,48,
%U A365656 16807,50,1331,169,14,1024,19683,390625,54,117649,80,14641,2197,28,15
%N A365656 Array T(n,k) read by antidiagonals (downward): T(n,1) = A005117(n) (squarefree numbers > 1); for k > 1, columns are nonsquarefree numbers (in descending order) with exactly the same prime factors as T(n,1).
%C A365656 Permutation of natural numbers.
%C A365656 Transpose of A284311 with a(0) = 1 prepended.
%C A365656 Essentially the same as A284457. - _R. J. Mathar_, Jan 23 2024
%H A365656 Michael De Vlieger, <a href="/A365656/b365656.txt">Table of n, a(n) for n = 0..11326</a> (rows 0..150, flattened)
%H A365656 Michael De Vlieger, <a href="/A365656/a365656.png">Log log scatterplot of log_10(a(n))</a>, n = 1..64981 (i.e., 360 rows, flattened).
%H A365656 Michael De Vlieger, <a href="/A365656/a365656_1.png">Log log scatterplot of log_10(a(n))</a>, n = 1..4096, showing primes in red, squarefree composites in green, composite prime powers in gold, and numbers neither squarefree nor prime powers in blue and purple; we show squareful numbers that are not prime powers in purple.
%F A365656 For prime n = p, T(p,k) = p^k.
%e A365656 Table T(n,k) for n = 1..12 and k = 1..6 shown below:
%e A365656   n\k |  1    2     3       4        5         6 ...
%e A365656   ----------------------------------------------
%e A365656    1  |  1
%e A365656    2  |  2    4     8      16       32        64
%e A365656    3  |  3    9    27      81      243       729
%e A365656    4  |  5   25   125     625     3125     15625
%e A365656    5  |  6   12    18      24       36        48
%e A365656    6  |  7   49   343    2401    16807    117649
%e A365656    7  | 10   20    40      50       80       100
%e A365656    8  | 11  121  1331   14641   161051   1771561
%e A365656    9  | 13  169  2197   28561   371293   4826809
%e A365656   10  | 14   28    56      98      112       196
%e A365656   11  | 15   45    75     135      225       375
%e A365656   12  | 17  289  4913   83521  1419857  24137569
%e A365656   ...
%e A365656 Triangle begins:
%e A365656    1;
%e A365656    2;
%e A365656    4,   3;
%e A365656    8,   9,   5;
%e A365656   16,  27,  25,  6;
%e A365656   32,  81, 125, 12,  7;
%e A365656   64, 243, 625, 18, 49, 10;
%e A365656  ...
%t A365656 f[n_, k_ : 1] := Block[{c = 0, s = Sign[k], m}, m = n + s;
%t A365656     While[c < Abs[k], While[! SquareFreeQ@ m, If[s < 0, m--, m++]];
%t A365656      If[s < 0, m--, m++]; c++];
%t A365656     m + If[s < 0, 1, -1] ] (* after Robert G.Wilson v at A005117 *);
%t A365656   T[n_, k_] := T[n, k] =
%t A365656     Which[And[n == 1, k == 1], 2, k == 1, f@T[n - 1, k],
%t A365656      PrimeQ@ T[n, 1], T[n, 1]^k, True,
%t A365656      Module[{j = T[n, k - 1]/T[n, 1] + 1},
%t A365656       While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; {1}~Join~
%t A365656    Table[T[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // TableForm
%Y A365656 Cf. A002260, A005117, A007947, A065642, A284311, A284457.
%K A365656 nonn,tabf
%O A365656 0,2
%A A365656 _Michael De Vlieger_, Nov 17 2023

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