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 A348142 #33 Jul 04 2025 21:53:32
%S A348142 1,6,3,60,78,9,120,7620,1014,21,360,28920,967740,12246,81,840,261720,
%T A348142 6969720,116136420,171366,147,3360,1422120
%N A348142 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in everyone of its p parts.
%C A348142 It appears that the first row is A318843 and that the first column is A250070.
%C A348142 Columns 1 and 2 both are identical with those of the table in A348171 and row 1 is identical with that of A348171.
%C A348142 In the remainder of the 7th antidiagonal a(24..26) > 120*10^6, a(27) = 1922622, and a(28) = 903.
%F A348142 a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.
%F A348142 T(w(n),p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2, w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).
%e A348142 The 10x8 section of the table T(w,p) with dashes indicating values greater than 120*10^6; rows w denote the common maximum width in all parts and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
%e A348142 w\p | 1 2 3 4 5 6 7 8 ...
%e A348142 --------------------------------------------------------------------------
%e A348142 1 | 1 3 9 21 81 147 729 903
%e A348142 2 | 6 78 1014 12246 171366 1922622 28960854 -
%e A348142 3 | 60 7620 967740 116136420 - - -
%e A348142 4 | 120 28920 6969720 -
%e A348142 5 | 360 261720 -
%e A348142 6 | 840 1422120 -
%e A348142 7 | 3360 22622880 -
%e A348142 8 | 2520 12728520 -
%e A348142 9 | 5040 50858640 -
%e A348142 10| 10080 -
%e A348142 ...
%e A348142 The symmetric representation of sigma for T(2,2) = 78 consists of the two parts (84, 84) of maximum widths (2, 2), and that of T(2,3) = 1014 consists of the three parts (1020, 156, 1020) of maximum widths (2, 2, 2).
%t A348142 (* function a341969 is defined in A341969 *)
%t A348142 a348142[n_, {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, Monitor[For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], #!=0&], #[[1]]!=0&]]; c=Length[s]; u=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]], list]; list]
%t A348142 table=a348142[120000000, {10, 10}] (* 10x10 table; very long computation time *)
%t A348142 p[n_] := n-row[n-1](row[n-1]+1)/2
%t A348142 w[n_] := row[n-1]-p[n]+2
%t A348142 Map[table[[w[#], p[#]]]&, Range[23]] (* sequence data *)
%Y A348142 Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348171.
%K A348142 nonn,tabl,more
%O A348142 1,2
%A A348142 _Hartmut F. W. Hoft_, Oct 04 2021