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A346969
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
1, 3, 5, 9, 7, 15, 21, 11, 25, 27, 63, 13, 35, 33, 81, 147, 17, 45, 39, 99, 171, 357, 19, 49, 51, 117, 189, 399, 903, 23, 77, 55, 153, 207, 441, 987, 2499, 29, 91, 57, 165, 243, 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
OFFSET
1,2
COMMENTS
This sequence is a permutation of the odd positive integers.
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).
EXAMPLE
The 10x10 initial submatrix of table T(n,k):
n\k | 2 3 4 5 6 7 8 9 10 11 ...
------------------------------------------------------------------
1 | 3 9 21 63 147 357 903 2499 6069 13915
2 | 5 15 27 81 171 399 987 2709 6321 14847
3 | 7 25 33 99 189 441 1029 2793 6325 15125
4 | 11 35 39 117 207 483 1113 2961 6783 15141
5 | 13 45 51 153 243 513 1197 3025 6875 15351
6 | 17 49 55 165 261 567 1239 3087 6909 15729
7 | 19 77 57 195 275 609 1265 3249 7011 16023
8 | 23 91 65 231 279 621 1281 3339 7203 16611
9 | 29 121 69 255 297 651 1375 3381 7353 16779
10| 31 135 75 273 333 729 1407 3591 7581 17157
...
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.
MATHEMATICA
(* function a341969 is defined in A341969 *)
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"]]
parts[n_] := n-row[n-1](row[n-1]+1)/2
rank[n_] := row[n-1]-parts[n]+2
a346969[sMatrix_, aD_] := Prepend[Map[sMatrix[[ rank[#], parts[#]]] &, Range[aD (aD + 1)/2]], 1] /; MatrixQ[sMatrix] && aD <= Length[sMatrix]
m40000=sArray[40000, 11] (* entire 11x11 table filled *)
a346969[m40000, 11] (* sequence data: 1 followed by the first 11 antidiagonals in the table *)
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Oct 06 2021
STATUS
approved