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.

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, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113

Offset: 1

Author

Omar E. Pol, Apr 06 2014

Keywords

• nonn
• tabl
• changed

Comments

This is a permutation of the positive integers.

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.

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

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

Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

Examples

			Array begins:
   1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
   2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
   4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
   6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
   8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
  12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
  16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
  18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
  20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
  24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
  ...
[Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]
		

Links

• Index entries for sequences that are permutations of the natural numbers

Crossrefs

For programs see A237271 and A237593.

Row 1 is A239663.

Column 1 is A174973.

Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.

For more information see A239663 and A239665.

Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.

Cf. A341969, A341970, A341971, A357581.

Programs

Mathematica

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
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]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)

Extensions

a(n) > 128 from Michel Marcus, Apr 08 2014