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A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).

%I A377667 #13 Feb 19 2025 12:20:13
%S A377667 60,10728,210,315,7620,810,495,1155,840456,2070,525,28158,945,88410,
%T A377667 7290,1275,1995,30555,1575,408150,12810,1287,2625,3003,22365,2835,
%U A377667 1313010,45450,6105,3315,10659,18975,382305,11385
%N A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).
%C A377667 When m is odd and SRS(m) has maximum width 3 then SRS(m) has at least 3 parts because the first and last parts of SRS(m) consist of a single leg of width 1. Therefore, the first two rows of the table contain only even numbers. The numbers in the third row appear to be odd and divisible by 15.
%e A377667 a(8) = T(3,2) = 1155 is the smallest example whose symmetric representation of sigma has 3 parts and 4 counts of width 3 in its width pattern.
%e A377667 Upper left hand section of table T(i, j) = m, numbers m <= 10^7, Columns j indicate 2j occurrences of width 3 in the width pattern of m. T(2, 7) > 10^7.
%e A377667 i\j| 1       2       3       4       5       6       7       8    ...
%e A377667 ---------------------------------------------------------------------
%e A377667 1  | 60      210     810     2070    7290    12810   45450   146610
%e A377667 2  | 10728   7620    840456  88410   408150  1313010 >10^7   8596710
%e A377667 3  | 315     1155    945     1575    2835    11385   8505    40095
%e A377667 4  | 495     28158   30555   22365   382305  296835  256095  199395
%e A377667 5  | 525     1995    3003    18975   15147   23925   14553   186219
%e A377667 6  | 1275    2625    10659   35217   132957  818363  312039  1760031
%e A377667 7  | 1287    3315    13125   37107   44289   195415  482937  258687
%e A377667 8  | 6105    3861    31875   65625   132153  149435  807495  1426113
%e A377667 9  | 3591    10773   56889   66861   254065  797979  319599  2199477
%e A377667 10 | 6783    16443   57477   222999  417175  1540875 768339  4670991
%e A377667 11 | 18963   35397   106191  965979  1025973 1770783 2489151 7547427
%e A377667 12 | 90801   58653   47481   1223365 2449785 4600617 ...     ...
%e A377667 13 | 152019  107457  817209  2213253 1740081 4310481
%e A377667 14 | 257397  297087  410571  3086349 3552213 5170055
%e A377667 15 | 335225  815409  1360989 2079609 ...     ...
%e A377667 16 | 1523319 2600283 1642557 2563239
%e A377667 17 | 1473725 1739375 4116777 ...
%e A377667 18 | 4008125 3826625 3687475
%e A377667 19 | 7576085 7937875 ...
%e A377667 ...  ...     ...
%t A377667 (* widthPattern[ ] and its support functions are defined in A376829 *)
%t A377667 t377667[b_, {r_, c_}] := Module[{t=ConstantArray[0, {r, c}], k, wP, c3, p3}, For[k=1, k<=b, k++, wP=widthPattern[k]; If[Max[wP]==3, c3=Count[wP, 3]; If[EvenQ[c3]&&c3<=2c, c3/=2; p3=Length[Select[SplitBy[wP, #!=0&], First[#]!=0&]]; If[p3<=r &&t[[p3, c3]]==0, t[[p3, c3]]=k]]]]; t]
%t A377667 t377667[1540875, {10, 6}] (* complete 10 x 6 upper left hand section of table *)
%Y A377667 Subsequence of A376829.
%Y A377667 Cf. A237591, A237593, A249223, A341969.
%K A377667 nonn,tabl,more
%O A377667 1,1
%A A377667 _Hartmut F. W. Hoft_, Nov 03 2024