Hartmut F. W. Hoft, <a href="/A253258/b253258_1.txt">Table of n, a(n) for n = 1..820</a>

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List of row position of the smallest odd number in each width column:

width 1 2 3 4 5 6 7 8 9 10 11 12

row 1 3 18 74 26 139 132 125 276 604 237 466

-----

width 13 14 15 16 17 18 19 20

row 555 1065 837 1027 2911 3095 1063 1623

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1, 6, 60, 120, 360, ...

2, 12, 72, ...

3, 15, 84, ...

4, 18, ...

5, 20, ...

7, ...

...

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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)

Hartmut F. W. Hoft, <a href="/A253258/b253258_1.txt">Table of n, a(n) for n = 1..820</a>

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

...

List of row position of the smallest odd number in each width column:

width 1 2 3 4 5 6 7 8 9 10 11 12

row 1 3 18 74 26 139 132 125 276 604 237 466

-----

width 13 14 15 16 17 18 19 20

row 555 1065 837 1027 2911 3095 1063 1623

(End)

(* 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 *)

Cf. A005279, A174905, A347980, A357581.

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