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A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

%I A240062 #81 Aug 28 2025 21:56:51
%S A240062 1,2,3,4,5,9,6,7,15,21,8,10,25,27,63,12,11,35,33,81,147,16,13,45,39,
%T A240062 99,171,357,18,14,49,51,117,189,399,903,20,17,50,55,153,207,441,987,
%U A240062 2499,24,19,70,57,165,243,483,1029,2709,6069,28,22,77,65,195,261,513,1113
%N A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.
%C A240062 This is a permutation of the positive integers.
%C A240062 All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
%C A240062 The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - _Omar E. Pol_, Oct 01 2018
%C A240062 Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - _Hartmut F. W. Hoft_, Oct 04 2022
%C A240062 Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - _Omar E. Pol_, Aug 25 2025
%H A240062 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A240062 Array begins:
%e A240062    1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
%e A240062    2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
%e A240062    4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
%e A240062    6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
%e A240062    8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
%e A240062   12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
%e A240062   16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
%e A240062   18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
%e A240062   20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
%e A240062   24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
%e A240062   ...
%e A240062 [Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]
%t A240062 (* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
%t A240062 partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
%t A240062 widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]<r, AppendTo[list[[parts]], k]]]; Transpose[PadRight[list, {c, r}, "..."]]]
%t A240062 a240062[n_, r_] := Module[{arr=widthTable[n, {r, r}], vec=Table[0, PolygonalNumber[r]], i, j}, For[i=1, i<=r, i++, For[j=r-i+1, j>=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
%t A240062 a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
%t A240062 a240062[6069, 10] (* data *)
%t A240062 a240062T[7581, 10] (* 10 X 10 array - _Hartmut F. W. Hoft_, Oct 04 2022 *)
%Y A240062 For programs see A237271 and A237593.
%Y A240062 Row 1 is A239663.
%Y A240062 Column 1 is A174973.
%Y A240062 Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.
%Y A240062 For more information see A239663 and A239665.
%Y A240062 Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.
%Y A240062 Cf. A341969, A341970, A341971, A357581.
%K A240062 nonn,tabl,changed
%O A240062 1,2
%A A240062 _Omar E. Pol_, Apr 06 2014
%E A240062 a(n) > 128 from _Michel Marcus_, Apr 08 2014