A348142 - OEIS

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.

%I #24 Nov 20 2021 21:08:38

%S 1,6,3,60,78,9,120,7620,1014,21,360,28920,967740,12246,81,840,261720,

%T 6969720,116136420,171366,147,3360,1422120

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

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

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

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

%F 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 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 w\p | 1 2 3 4 5 6 7 8 ...

%e --------------------------------------------------------------------------

%e 1 | 1 3 9 21 81 147 729 903

%e 2 | 6 78 1014 12246 171366 1922622 28960854 -

%e 3 | 60 7620 967740 116136420 - - -

%e 4 | 120 28920 6969720 -

%e 5 | 360 261720 -

%e 6 | 840 1422120 -

%e 7 | 3360 22622880 -

%e 8 | 2520 12728520 -

%e 9 | 5040 50858640 -

%e 10| 10080 -

%e ...

%e 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 (* function a341969 is defined in A341969 *)

%t 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]

%t table=a348142[12000000}, {10, 10}] (* 10x10 table; very long computation time *)

%t p[n_] := n-row[n-1](row[n-1]+1)/2

%t w[n_] := row[n-1]-p[n]+2

%t Map[table[[w[#], p[#]]]&, Range[23]] (* sequence data *)

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

%K nonn,tabl,more

%O 1,2

%A _Hartmut F. W. Hoft_, Oct 04 2021