This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347285 #17 Sep 08 2021 09:52:43
%S A347285 0,1,2,1,3,1,4,2,1,5,3,2,1,6,3,2,1,7,4,2,1,8,5,3,2,1,9,5,3,2,1,10,6,4,
%T A347285 3,2,1,11,6,4,3,2,1,12,7,4,3,2,1,13,8,5,4,3,2,1,14,8,5,4,3,2,1,15,9,6,
%U A347285 4,3,2,1,16,10,6,4,3,2,1,17,10,6,4,3,2,1
%N A347285 Irregular triangle T(n,k) starting with n followed by e_k = floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.
%C A347285 Irregular triangle T(n,k) starting with n followed by e_k corresponding to the largest 1 < p_k^e_k < p_(k-1)^e_(k-1).
%C A347285 T(0,1) = 0 by convention; 0 is not allowed for n > 0.
%C A347285 T(n,k) > T(n,k+1). The least first difference among row n is 1.
%C A347285 Conjecture: let S be the sum of the absolute values of the first differences of terms in row n. For all n > 0, n - S = 1. - _Michael De Vlieger_, Aug 27 2021
%H A347285 Michael De Vlieger, <a href="/A347285/b347285.txt">Table of n, a(n) for n = 0..10367</a> (rows 0 <= n <= 300, flattened)
%F A347285 T(n,1) = n; T(n,k) = floor(log_p_k(p_(k-1)^T(n,k-1))).
%F A347285 A000217(A089576(n)) <= sum of terms in row n.
%e A347285 Row 0 contains {0} by convention.
%e A347285 Row 1 contains {1} since we can find no nonzero exponent e such that 3^e < 2^1.
%e A347285 Row 2 contains {2,1} 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 A347285 Row 3 contains {3,1} since 2^3 > 3^1.
%e A347285 Row 4 contains {4,2,1} since 2^4 > 3^2 > 5^1, etc.
%e A347285 Triangle begins:
%e A347285 0
%e A347285 1
%e A347285 2 1
%e A347285 3 1
%e A347285 4 2 1
%e A347285 5 3 2 1
%e A347285 6 3 2 1
%e A347285 7 4 2 1
%e A347285 8 5 3 2 1
%e A347285 9 5 3 2 1
%e A347285 10 6 4 3 2 1
%e A347285 11 6 4 3 2 1
%e A347285 12 7 4 3 2 1
%e A347285 13 8 5 4 3 2 1
%e A347285 14 8 5 4 3 2 1
%e A347285 15 9 6 4 3 2 1
%e A347285 16 10 6 4 3 2 1
%e A347285 ...
%t A347285 Array[NestWhile[Block[{p = Prime[#2]}, Append[#1, {p^#, #} &@ Floor@ Log[p, #1[[-1, 1]]]]] & @@ {#, Length@ # + 1} &, {{2^#, #}}, #[[-1, -1]] > 1 &][[All, -1]] &, 18, 0] // Flatten
%Y A347285 Cf. A000217, A000961, A089576 (row lengths).
%K A347285 nonn,tabf,easy
%O A347285 0,3
%A A347285 _Michael De Vlieger_, Aug 26 2021