A347288 Irregular triangle T(n,k) starting with 2^n followed by p_k^e_k = p_k^floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.

1, 2, 4, 3, 8, 3, 16, 9, 5, 32, 27, 25, 7, 64, 27, 25, 7, 128, 81, 25, 7, 256, 243, 125, 49, 11, 512, 243, 125, 49, 11, 1024, 729, 625, 343, 121, 13, 2048, 729, 625, 343, 121, 13, 4096, 2187, 625, 343, 121, 13, 8192, 6561, 3125, 2401, 1331, 169, 17

Offset: 0

Author

Michael De Vlieger, Aug 28 2021

Keywords

• nonn
• tabf
• easy

Comments

T(0,1) = 1 by convention.

T(n,1) = 2^n. T(n,k) = p_k^e_k such that p_k^T(n,k) is the largest 1 < p_k^e_k < p_(k-1)^e_(k-1).

Examples

			Row 0 contains {1} by convention.
Row 1 contains {2} since no nonzero exponent e exists such that 3^e < 2^1.
Row 2 contains {4,3} since 3^1 < 2^2 yet 3^2 > 2^2. (We assume hereinafter that the powers listed are the largest possible smaller than the immediately previous term.)
Row 3 contains {8,3} since 2^3 > 3^1.
Row 4 contains {16,9,5} since 2^4 > 3^2 > 5^1, etc.
Triangle begins:
           2      3      5      7     11    13    17  ...
  --------------------------------------------------
  0:       1
  1:       2
  2:       4      3
  3:       8      3
  4:      16      9      5
  5:      32     27     25      7
  6:      64     27     25      7
  7:     128     81     25      7
  8:     256    243    125     49     11
  9:     512    243    125     49     11
  10:   1024    729    625    343    121    13
  11:   2048    729    625    343    121    13
  12:   4096   2187    625    343    121    13
  13:   8192   6561   3125   2401   1331   169   17
  14:  16384   6561   3125   2401   1331   169   17
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10367 (rows 0 <= n <= 300, flattened)

Crossrefs

Cf. A000217, A000961, A089576, A347284.

Programs

Mathematica

{{1}}~Join~Array[Most@ NestWhile[Block[{p = Prime[#2]}, Append[#1, p^Floor@ Log[p, #1[[-1]]]]] & @@ {#, Length@ # + 1} &, {2^#}, #[[-1]] > 1 &] &, 13] (* Michael De Vlieger, Aug 28 2021 *)

Formula

T(n,1) = 2^n; T(n,k) = p_k^floor(log_p_k(p_(k-1)^T(n,k-1))).

A347385(n,k) = p_k^T(n,k).

A089576(n) = row lengths.

A347284(n) = product of row n.