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A348171 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 at least one of its p parts.

%I A348171 #23 Apr 13 2025 01:46:42
%S A348171 1,6,3,60,78,9,120,7620,15,21,360,28920,315,75,81,840,261720,1326,495,
%T A348171 63,147,3360,1422120,3465,22542,525,189,729,2520,22622880,17325,44574,
%U A348171 5005,1275,357,903,5040,12728520,45045,199578,6435,16575,1287,1197,3025,10080,50858640,51975,7734558,34034,131835,2145,3861,2499,6875
%N A348171 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 at least one of its p parts.
%C A348171 The first row of the table below is A318843 and the first column is A250070.
%C A348171 T(1,k+1) <= 3^k, for all k>=0, since for k=2j the (j+1)-st part in the symmetric representation of sigma(3^k) extends across the diagonal, and for k=2j+1 the (j+1)-st part is completed before the diagonal.
%C A348171 The data computed so far for a partially filled table of 15 rows and 15 columns, show that all rows, all columns (except column 4 for n <= 6 *10^7), and the diagonal are nonmonotonic.
%F A348171 a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.
%F A348171 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 A348171 The 10x10 section of the table with dashes indicating values greater than 6*10^7; rows w denote the maximum width and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
%e A348171 w\p | 1     2        3      4       5       6       7       8        9   ...
%e A348171 ----------------------------------------------------------------------------
%e A348171   1 | 1     3        9      21      81      147     729     903      3025
%e A348171   2 | 6     78       15     75      63      189     357     1197     2499
%e A348171   3 | 60    7620     315    495     525     1275    1287    3861     3591
%e A348171   4 | 120   28920    1326   22542   5005    16575   2145    29325    11583
%e A348171   5 | 360   261720   3465   44574   6435    131835  76125   24225    82593
%e A348171   6 | 840   1422120  17325  199578  34034   83655   196707  468027   62985
%e A348171   7 | 3360  22622880 45045  7734558 153153  442442  314925  1108965  471975
%e A348171   8 | 2520  12728520 51975     -    205275  2067065 1429275 2359875  557175
%e A348171   9 | 5040  50858640 225225    -    646646  2863718 2395197 5353725  2785875
%e A348171   10| 10080    -     405405    -    1990989 2124694 6500375 36535499 7753875
%e A348171    ...
%e A348171 The symmetric representation of sigma for T(2,3) = 15 consists of the three parts (8, 8, 8) of maximum widths (1, 2, 1), and that of T(3,3) = 315 consists of the three parts (158, 308, 158) of maximum widths (1, 3, 1).
%t A348171 (* function a341969 is defined in A341969 *)
%t A348171 a348171[n_,  {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], # != 0 &], #[[1]] != 0 &]]; c = Length[s]; u = Max[s]; If[u<=w && c<=p, If[list[[u, c]] == 0, list[[u, c]] = k ]]]; list]
%t A348171 table=a348171[60000000, {15, 15}] (* 15x15 table; very long computation time *)
%t A348171 p[n_] := n-row[n-1](row[n-1]+1)/2
%t A348171 w[n_] := row[n-1]-p[n]+2
%t A348171 Map[table[[w[#], p[#]]]&, Range[55]] (* sequence data *)
%Y A348171 Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348142.
%K A348171 nonn,tabl
%O A348171 1,2
%A A348171 _Hartmut F. W. Hoft_, Oct 04 2021