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1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 parts.
2

%I #66 Oct 07 2021 15:02:28

%S 1,3,5,9,7,15,21,11,25,27,63,13,35,33,81,147,17,45,39,99,171,357,19,

%T 49,51,117,189,399,903,23,77,55,153,207,441,987,2499,29,91,57,165,243,

%U 483,1029,2709,6069,31,121,65,195,261,513,1113,2793,6321,13915,37,135,69,231,275,567,1197,2961,6325,14847,29095

%N 1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 parts.

%C This sequence is a permutation of the odd positive integers.

%C The first row of table T(n,k) preceded by a(1) = 1 is A239663; the first column is the sequence A065091 of odd primes; the second column contains the squares of the odd primes as a subsequence (see also A247687).

%e The 10x10 initial submatrix of table T(n,k):

%e n\k | 2 3 4 5 6 7 8 9 10 11 ...

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

%e 1 | 3 9 21 63 147 357 903 2499 6069 13915

%e 2 | 5 15 27 81 171 399 987 2709 6321 14847

%e 3 | 7 25 33 99 189 441 1029 2793 6325 15125

%e 4 | 11 35 39 117 207 483 1113 2961 6783 15141

%e 5 | 13 45 51 153 243 513 1197 3025 6875 15351

%e 6 | 17 49 55 165 261 567 1239 3087 6909 15729

%e 7 | 19 77 57 195 275 609 1265 3249 7011 16023

%e 8 | 23 91 65 231 279 621 1281 3339 7203 16611

%e 9 | 29 121 69 255 297 651 1375 3381 7353 16779

%e 10| 31 135 75 273 333 729 1407 3591 7581 17157

%e ...

%e a(9) = 25 = T(3,3) since only 9 and 15 are smaller odd numbers whose symmetric representation of sigma consists of three parts. All 3 parts of the symmetric representation of sigma for 9 and for 25 have width 1 while the center part for that of 15 has width 2.

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

%t sArray[b_, pMax_] := Module[{list=Table[{}, pMax+1], i, p}, For[i=3, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax+1&&Length[list[[p]]]<pMax, AppendTo[list[[p]], i]]]; If[MatrixQ[Rest[list]], Transpose[Rest[list]], "square matrix not filled"]]

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

%t rank[n_] := row[n-1]-parts[n]+2

%t a346969[sMatrix_, aD_] := Prepend[Map[sMatrix[[ rank[#], parts[#]]] &, Range[aD (aD + 1)/2]], 1] /; MatrixQ[sMatrix] && aD <= Length[sMatrix]

%t m40000=sArray[40000, 11] (* entire 11x11 table filled *)

%t a346969[m40000, 11] (* sequence data: 1 followed by the first 11 antidiagonals in the table *)

%Y Cf. A065091, A237270, A237271, A237593, A239663, A247687, A320537, A341969, A341970, A341971, A348171.

%K nonn,tabf

%O 1,2

%A _Hartmut F. W. Hoft_, Oct 06 2021