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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.

%I A347288 #16 Jun 30 2025 04:26:50
%S A347288 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,
%T A347288 11,512,243,125,49,11,1024,729,625,343,121,13,2048,729,625,343,121,13,
%U A347288 4096,2187,625,343,121,13,8192,6561,3125,2401,1331,169,17
%N 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.
%C A347288 T(0,1) = 1 by convention.
%C A347288 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).
%H A347288 Michael De Vlieger, <a href="/A347288/b347288.txt">Table of n, a(n) for n = 0..10367</a> (rows 0 <= n <= 300, flattened)
%F A347288 T(n,1) = 2^n; T(n,k) = p_k^floor(log_p_k(p_(k-1)^T(n,k-1))).
%F A347288 A347385(n,k) = p_k^T(n,k).
%F A347288 A089576(n) = row lengths.
%F A347288 A347284(n) = product of row n.
%e A347288 Row 0 contains {1} by convention.
%e A347288 Row 1 contains {2} since no nonzero exponent e exists such that 3^e < 2^1.
%e A347288 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.)
%e A347288 Row 3 contains {8,3} since 2^3 > 3^1.
%e A347288 Row 4 contains {16,9,5} since 2^4 > 3^2 > 5^1, etc.
%e A347288 Triangle begins:
%e A347288            2      3      5      7     11    13    17  ...
%e A347288   --------------------------------------------------
%e A347288   0:       1
%e A347288   1:       2
%e A347288   2:       4      3
%e A347288   3:       8      3
%e A347288   4:      16      9      5
%e A347288   5:      32     27     25      7
%e A347288   6:      64     27     25      7
%e A347288   7:     128     81     25      7
%e A347288   8:     256    243    125     49     11
%e A347288   9:     512    243    125     49     11
%e A347288   10:   1024    729    625    343    121    13
%e A347288   11:   2048    729    625    343    121    13
%e A347288   12:   4096   2187    625    343    121    13
%e A347288   13:   8192   6561   3125   2401   1331   169   17
%e A347288   14:  16384   6561   3125   2401   1331   169   17
%e A347288   ...
%t A347288 {{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 *)
%Y A347288 Cf. A000217, A000961, A089576, A347284.
%K A347288 nonn,tabf,easy
%O A347288 0,2
%A A347288 _Michael De Vlieger_, Aug 28 2021