A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

1, 2, 6, 3, 12, 60, 4, 15, 72, 120, 5, 18, 84, 180, 360, 7, 20, 90, 240, 420, 840, 8, 24, 126, 252, 720, 1080, 3360, 9, 28, 140, 336, 1008, 1260, 3600, 2520, 10, 30, 144, 378, 1200, 1440, 3780, 5544, 5040, 11, 35, 168, 432, 1320, 1680, 3960, 6300, 7560, 10080, 13, 36, 198, 480, 1512, 1800, 4200, 6720, 9240, 12600, 15120

Offset: 1

Author

Omar E. Pol, Jul 08 2015

Keywords

• nonn
• tabl

Comments

This is a permutation of the natural numbers.

Row 1 gives A250070.

For more information about the widths of the symmetric representation of sigma see A249351 and A250068.

The next term: 120 < T(2,4) < 360.

From Hartmut F. W. Hoft, Sep 20 2024: (Start)

Column T(j,1), j>=1, forms A174905 and is a permutation of A357581. Numbers T(j,k), j>=1 and k>1, form A005279. Conjecture: Every column of the square array contains odd numbers.

The sequence of smallest odd numbers in each column forms A347980. E.g., in column 12 the smallest odd number is T(466, 12) = 765765 = A347980(12) which is equivalent to A250068(765765) = 12. (End)

Examples

			The corner of the square array T(j,k) begins:
  1,  6, 60, 120, 360, ...
  2, 12, 72, ...
  3, 15, 84, ...
  4, 18, ...
  5, 20, ...
  7, ...
  ...
For j = 1 and k = 2; T(1,2) is the first number n such that the symmetric representation of sigma(n) has a part with maximum width 2 as shown below:
.
      Dyck paths            Cells              Widths
      _ _ _ _             _ _ _ _
      _ _ _  |_          |_|_|_|_|_          / / / /
           |   |_              |_|_|_              / /
           |_ _  |             |_|_|_|             / / /
               | |                 |_|                 /
               | |                 |_|                 /
               | |                 |_|                 /
.
The widths of the symmetric representation of sigma(6) = 12 are [1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1], also the 6th row of triangle A249351.
From _Hartmut F. W. Hoft_, Sep 20 2024: (Start)
Extending the terms T(j,k) to a 12x12 square array:
j\k 1  2  3   4   5    6    7    8     9     10    11    12
--------------------------------------------------------------
1 | 1  6  60  120 360  840  3360 2520  5040  10080 15120 32760
2 | 2  12 72  180 420  1080 3600 5544  7560  12600 20160 36960
3 | 3  15 84  240 720  1260 3780 6300  9240  13860 25200 39600
4 | 4  18 90  252 1008 1440 3960 6720  10920 15840 35280 41580
5 | 5  20 126 336 1200 1680 4200 6930  11880 16380 40320 43680
6 | 7  24 140 378 1320 1800 4320 7140  14040 16800 42840 45360
7 | 8  28 144 432 1512 1980 4620 7920  16632 18480 46800 46200
8 | 9  30 168 480 1560 2016 4680 8190  17160 18900 47880 47520
9 | 10 35 198 504 1848 2100 5280 8400  17640 21420 56160 49140
10| 11 36 210 540 1890 2160 5400 9360  18720 21840 56700 51480
11| 13 40 216 594 2184 2340 5460 10296 19800 22680 57120 52920
12| 14 42 264 600 2310 2640 5940 10800 20790 23760 57960 54600
...
(End)
		

Links

• Hartmut F. W. Hoft, Table of n, a(n) for n = 1..820
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A250071.

Cf. A005279, A174905, A347980, A357581.

Programs

Mathematica

(* Computing table T(j,k) of size mxn with bound b *)
eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mWidth[n_] :=Max[FoldList[#1+If[OddQ[#2], 1, -1]&, sDiv[n]]]
t253258[{m_, n_}, b_] := Module[{s=Table[0, {i, m+1}, {j, n}], k=1, w, f}, While[k<=b, w=mWidth[k]; If[w<=n, f=s[[m+1, w]]; If[fHartmut F. W. Hoft, Sep 20 2024 *)

Extensions

More terms from Charlie Neder, Jan 11 2019