OFFSET
0,2
COMMENTS
Permutation of natural numbers.
Transpose of A284311 with a(0) = 1 prepended.
Essentially the same as A284457. - R. J. Mathar, Jan 23 2024
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.
FORMULA
For prime n = p, T(p,k) = p^k.
EXAMPLE
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;
...
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Michael De Vlieger, Nov 17 2023
STATUS
approved