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.

1, 6, 3, 60, 78, 9, 120, 7620, 1014, 21, 360, 28920, 967740, 12246, 81, 840, 261720, 6969720, 116136420, 171366, 147, 3360, 1422120

Offset

1,2

Comments

It appears that the first row is A318843 and that the first column is A250070.

Columns 1 and 2 both are identical with those of the table in A348171 and row 1 is identical with that of A348171.

In the remainder of the 7th antidiagonal a(24..26) > 120*10^6, a(27) = 1922622, and a(28) = 903.

Links

Table of n, a(n) for n=1..23.

Formula

a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.

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).

Example

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)).
w\p | 1     2         3        4         5        6        7        8  ...
--------------------------------------------------------------------------
  1 | 1     3         9        21        81       147     729      903
  2 | 6     78        1014     12246     171366   1922622 28960854  -
  3 | 60    7620      967740   116136420   -         -       -
  4 | 120   28920     6969720    -
  5 | 360   261720      -
  6 | 840   1422120     -
  7 | 3360  22622880    -
  8 | 2520  12728520    -
  9 | 5040  50858640    -
  10| 10080    -
   ...
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).

Mathematica

(* function a341969 is defined in A341969 *)
a348142[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=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]]; list]
table=a348142[12000000}, {10, 10}] (* 10x10 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&,  Range[23]] (* sequence data *)

Crossrefs

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348171.

Sequence in context: A286782 A054380 A348171 * A324673 A367377 A324004

Adjacent sequences: A348139 A348140 A348141 * A348143 A348144 A348145

Keyword

nonn,tabl,more

Author

Hartmut F. W. Hoft, Oct 04 2021

Status

approved