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).

1, 2, 4, 3, 8, 9, 5, 16, 27, 25, 6, 32, 81, 125, 12, 7, 64, 243, 625, 18, 49, 10, 128, 729, 3125, 24, 343, 20, 11, 256, 2187, 15625, 36, 2401, 40, 121, 13, 512, 6561, 78125, 48, 16807, 50, 1331, 169, 14, 1024, 19683, 390625, 54, 117649, 80, 14641, 2197, 28, 15

Offset: 0

Author

Michael De Vlieger, Nov 17 2023

Keywords

• nonn
• tabf

Comments

Permutation of natural numbers.

Transpose of A284311 with a(0) = 1 prepended.

Essentially the same as A284457. - R. J. Mathar, Jan 23 2024

Examples

			Table T(n,k) for n = 1..12 and k = 1..6 shown below:
  n\k |  1    2     3       4        5         6 ...
  ----------------------------------------------
   1  |  1
   2  |  2    4     8      16       32        64
   3  |  3    9    27      81      243       729
   4  |  5   25   125     625     3125     15625
   5  |  6   12    18      24       36        48
   6  |  7   49   343    2401    16807    117649
   7  | 10   20    40      50       80       100
   8  | 11  121  1331   14641   161051   1771561
   9  | 13  169  2197   28561   371293   4826809
  10  | 14   28    56      98      112       196
  11  | 15   45    75     135      225       375
  12  | 17  289  4913   83521  1419857  24137569
  ...
Triangle begins:
   1;
   2;
   4,   3;
   8,   9,   5;
  16,  27,  25,  6;
  32,  81, 125, 12,  7;
  64, 243, 625, 18, 49, 10;
 ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11326 (rows 0..150, flattened)
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..64981 (i.e., 360 rows, flattened).
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), 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.

Crossrefs

Cf. A002260, A005117, A007947, A065642, A284311, A284457.

Programs

Mathematica

f[n_, k_ : 1] := Block[{c = 0, s = Sign[k], m}, m = n + s;
    While[c < Abs[k], While[! SquareFreeQ@ m, If[s < 0, m--, m++]];
     If[s < 0, m--, m++]; c++];
    m + If[s < 0, 1, -1] ] (* after Robert G.Wilson v at A005117 *);
  T[n_, k_] := T[n, k] =
    Which[And[n == 1, k == 1], 2, k == 1, f@T[n - 1, k],
     PrimeQ@ T[n, 1], T[n, 1]^k, True,
     Module[{j = T[n, k - 1]/T[n, 1] + 1},
      While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; {1}~Join~
   Table[T[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // TableForm

Formula

For prime n = p, T(p,k) = p^k.