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A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

%I A320537 #35 Nov 03 2021 22:21:44
%S A320537 2,4,10,6,14,50,8,22,70,230,12,26,98,250,1150,16,34,110,290,1250,5050,
%T A320537 18,38,130,310,1450,5150,22310,20,44,154,370,1550,5290,23230,106030,
%U A320537 24,46,170,406,1850,5350,23690,106490,510050,28,52,182,410,2030,5450,24610,107410,513130,2065450
%N A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.
%C A320537 This is a permutation of the positive even numbers (A299174).
%C A320537 The union of all odd-indexed columns gives A319796, the even numbers in A071562.
%C A320537 The union of all even-indexed columns gives A319802, the even numbers in A071561.
%e A320537 From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
%e A320537 The 10x10 section of table T(n,k):
%e A320537 (Table with first 20 terms from _Omar E. Pol_)
%e A320537 ------------------------------------------------------------------
%e A320537 n\k | 1   2   3    4    5     6     7      8       9       10  ...
%e A320537 ------------------------------------------------------------------
%e A320537   1 | 2   10  50   230  1150  5050  22310  106030  510050  2065450
%e A320537   2 | 4   14  70   250  1250  5150  23230  106490  513130  2115950
%e A320537   3 | 6   22  98   290  1450  5290  23690  107410  520150  2126050
%e A320537   4 | 8   26  110  310  1550  5350  24610  110170  530150  2157850
%e A320537   5 | 12  34  130  370  1850  5450  25070  112010  530450  2164070
%e A320537   6 | 16  38  154  406  2030  5650  25250  112930  532450  2168150
%e A320537   7 | 18  44  170  410  2050  5750  25750  114770  534290  2176550
%e A320537   8 | 20  46  182  430  2150  6250  25990  115690  537050  2186650
%e A320537   9 | 24  52  190  434  2170  6350  26450  116150  540350  2216950
%e A320537   10| 28  58  238  470  2350  6550  26750  117070  544870  2219650
%e A320537    ... (End)
%t A320537 (* function a341969 is defined in A341969 *)
%t A320537 sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]<pMax, AppendTo[list[[p]], i]]]; list]
%t A320537 rank[n_] := n-row[n-1](row[n-1]+1)/2
%t A320537 parts[n_] := row[n-1]-rank[n]+2
%t A320537 a320537[sMatrix_, aD_] := Map[sMatrix[[rank[#], parts[#]]]&, Range[aD (aD+1)/2]]/; MatrixQ[sMatrix]&&aD<=Length[sMatrix]
%t A320537 m2500000=sArray[2500000, 10] (* entire 10x10 matrix needs to be computed *)
%t A320537 a320537[m2500000, 10] (* Sequence Data a(1..55) *)
%t A320537 (* _Hartmut F. W. Hoft_, Oct 06 2021 *)
%Y A320537 Row 1 is A320521.
%Y A320537 Column 1 gives A174973 = A238443, without the 1.
%Y A320537 Column 2 gives A244894.
%Y A320537 Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.
%K A320537 nonn,tabl
%O A320537 1,1
%A A320537 _Omar E. Pol_, Oct 15 2018
%E A320537 Terms a(21) and beyond from _Hartmut F. W. Hoft_, Oct 06 2021