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URL: https://oeis.org/A347288

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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.
2
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
0,2
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).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10367 (rows 0 <= n <= 300, flattened)
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.
EXAMPLE
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
...
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 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Michael De Vlieger, Aug 28 2021
STATUS
approved