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 A342596 #10 Mar 18 2021 23:52:41
%S A342596 1,3,6,9,18,72,21,15,78,30,60,120,81,162,648,1296,5184,147,63,75,45,
%T A342596 1014,666,150,90,10728,3816,300,180,27744,504,360,1440,729,1458,5832,
%U A342596 11664,46656,93312,373248,903,357,189,231,465,165,105,135,1001,770,12246,4134,1482,1326,1830,690,390,858,210,378
%N A342596 Numbers k of the earliest occurrence of widths patterns in the symmetric representation of sigma listed in the ordering of patterns in A342595.
%C A342596 This sequence is the companion to A342595 in that a(n) is the smallest number k that has row n of the table in A342595 as its width pattern in the symmetric representation of sigma(k).
%C A342596 The number of possible width patterns of length n occurring up to the diagonal in symmetric representations of sigma is A001405(n). Those are realized for n <= 4. For larger n the actual number of width patterns is smaller. Only p symmetric patterns of length 2p-1 are realizable when a number has p odd divisors and p is prime. Patterns such as 1 0 1 2 3 ... k-1 k k-1 ... 3 2 1 0 1, k >= 4, i.e., numbers with at least 6 odd divisors, cannot be realized as width patterns in the symmetric representation of sigma. If n = 2^s * p * q^2, s >= 0, p < q odd primes, then 2^(s+1) < p and row(n) < 2^(s+1) * p must hold which leads to the contradiction q^2 < p^2; if n = 2^s * p^2 * q, s >= 0, p < q odd primes, then again 2^(s+1) < p and row(n) < 2^(s+1) * p must hold which leads to the contradiction p * q < p^2.
%e A342596 a(17) = 5184 = 2^6 * 3^4 is the smallest number with width pattern (1 2 3 4 5 4 3 2 1).
%e A342596 a(18) = 147 = 3 * 7^2 is the smallest number with width pattern (1 0 1 0 1 0 1 0 1 0 1).
%t A342596 (* a341969[] defined in A341969 and lexicographicOrder[] in A342595 *)
%t A342596 a342596[n_] := Module[{listW={}, listK={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; Flatten[Map[First, Sort[Transpose[{listK, listW}], lexicographicOrder]]]]
%t A342596 Take[a342596[500000], 60]
%Y A342596 Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342595.
%K A342596 nonn
%O A342596 1,2
%A A342596 _Hartmut F. W. Hoft_, Mar 16 2021