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%I A253258 #50 Oct 23 2024 18:34:41
%S A253258 1,2,6,3,12,60,4,15,72,120,5,18,84,180,360,7,20,90,240,420,840,8,24,
%T A253258 126,252,720,1080,3360,9,28,140,336,1008,1260,3600,2520,10,30,144,378,
%U A253258 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
%N 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.
%C A253258 This is a permutation of the natural numbers.
%C A253258 Row 1 gives A250070.
%C A253258 For more information about the widths of the symmetric representation of sigma see A249351 and A250068.
%C A253258 The next term: 120 < T(2,4) < 360.
%C A253258 From _Hartmut F. W. Hoft_, Sep 20 2024: (Start)
%C A253258 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.
%C A253258 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)
%H A253258 Hartmut F. W. Hoft, <a href="/A253258/b253258.txt">Table of n, a(n) for n = 1..820</a>
%H A253258 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A253258 The corner of the square array T(j,k) begins:
%e A253258   1,  6, 60, 120, 360, ...
%e A253258   2, 12, 72, ...
%e A253258   3, 15, 84, ...
%e A253258   4, 18, ...
%e A253258   5, 20, ...
%e A253258   7, ...
%e A253258   ...
%e A253258 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:
%e A253258 .
%e A253258       Dyck paths            Cells              Widths
%e A253258       _ _ _ _             _ _ _ _
%e A253258       _ _ _  |_          |_|_|_|_|_          / / / /
%e A253258            |   |_              |_|_|_              / /
%e A253258            |_ _  |             |_|_|_|             / / /
%e A253258                | |                 |_|                 /
%e A253258                | |                 |_|                 /
%e A253258                | |                 |_|                 /
%e A253258 .
%e A253258 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.
%e A253258 From _Hartmut F. W. Hoft_, Sep 20 2024: (Start)
%e A253258 Extending the terms T(j,k) to a 12x12 square array:
%e A253258 j\k 1  2  3   4   5    6    7    8     9     10    11    12
%e A253258 --------------------------------------------------------------
%e A253258 1 | 1  6  60  120 360  840  3360 2520  5040  10080 15120 32760
%e A253258 2 | 2  12 72  180 420  1080 3600 5544  7560  12600 20160 36960
%e A253258 3 | 3  15 84  240 720  1260 3780 6300  9240  13860 25200 39600
%e A253258 4 | 4  18 90  252 1008 1440 3960 6720  10920 15840 35280 41580
%e A253258 5 | 5  20 126 336 1200 1680 4200 6930  11880 16380 40320 43680
%e A253258 6 | 7  24 140 378 1320 1800 4320 7140  14040 16800 42840 45360
%e A253258 7 | 8  28 144 432 1512 1980 4620 7920  16632 18480 46800 46200
%e A253258 8 | 9  30 168 480 1560 2016 4680 8190  17160 18900 47880 47520
%e A253258 9 | 10 35 198 504 1848 2100 5280 8400  17640 21420 56160 49140
%e A253258 10| 11 36 210 540 1890 2160 5400 9360  18720 21840 56700 51480
%e A253258 11| 13 40 216 594 2184 2340 5460 10296 19800 22680 57120 52920
%e A253258 12| 14 42 264 600 2310 2640 5940 10800 20790 23760 57960 54600
%e A253258 ...
%e A253258 (End)
%t A253258 (* Computing table T(j,k) of size mxn with bound b *)
%t A253258 eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
%t A253258 sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
%t A253258 mWidth[n_] :=Max[FoldList[#1+If[OddQ[#2], 1, -1]&, sDiv[n]]]
%t A253258 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]]
%t A253258 t253258[{12, 12}, 60000] (* _Hartmut F. W. Hoft_, Sep 20 2024 *)
%Y A253258 Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A250071.
%Y A253258 Cf. A005279, A174905, A347980, A357581.
%K A253258 nonn,tabl
%O A253258 1,2
%A A253258 _Omar E. Pol_, Jul 08 2015
%E A253258 More terms from _Charlie Neder_, Jan 11 2019

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