A253258 - OEIS

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

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OFFSET

1,2

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)

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

EXAMPLE

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)

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[f<m, s[[f+1, w]]=k; s[[m+1, w]]=f+1]]; k++]; Most[s]]

t253258[{12, 12}, 60000] (* Hartmut F. W. Hoft, Sep 20 2024 *)

CROSSREFS

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

Cf. A005279, A174905, A347980, A357581.

Sequence in context: A054619 A054618 A120859 * A098810 A360006 A081469

Adjacent sequences: A253255 A253256 A253257 * A253259 A253260 A253261

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Jul 08 2015

EXTENSIONS

More terms from Charlie Neder, Jan 11 2019

STATUS

approved