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A367377 Square array T(n, k), n >= 1, k >= 1, read by antidiagonals, of the least numbers whose symmetric representation of sigma instantiate the unimodal width pattern 1, 2, ..., n, ..., 2, 1 repeated k times separated by instances of width 0.

%I A367377 #22 Dec 06 2023 14:21:37
%S A367377 1,6,3,72,78,9,120,10728,1014,21,5184,28920,1598472,12246,81,1440,
%T A367377 53752896,6969720,230297976,171366,147,373248,4157280
%N A367377 Square array T(n, k), n >= 1, k >= 1, read by antidiagonals, of the least numbers whose symmetric representation of sigma instantiate the unimodal width pattern 1, 2, ..., n, ..., 2, 1 repeated k times separated by instances of width 0.
%C A367377 The numbers T(n, 1) instantiating a single unimodal pattern of width n form A250071(n). This first column is not increasing since T(5, 1) = 5184 > 1440 = T(6, 1).
%C A367377 The numbers T(1, k) instantiating the repeating unimodal patterns 1, 1, 0, 1, ..., 1, 0, 1, 0, ..., 0, 1, 0, 1, ... of width 1 form A318843(k). This first row is not increasing since T(1, 11) = 59049 > 29095 = T(1, 12).
%C A367377 The rows in the table are infinite since the numbers T(n, 1) * p^(k-1) >= T(n, k), with p the smallest prime greater than 2 * T(n, 1), instantiate the width pattern for T(n, k), though equality need not hold, as T(1, 4) = 21 = 3 * 7 < 1 * 3^3 = 27 demonstrates.
%C A367377 Conjecture 1: None of the rows and columns are increasing.
%C A367377 Conjecture 2: T(n, p) = T(n, 1) * A151800(2*T(n, 1))^(p-1) for n >= 1 and primes p.
%C A367377 Conjecture 3: T(p, q), p and q primes, is a record for its upper left hand rectangle in the table. Only one prime number index generally is not sufficient as the inequality 4157280 = T(6, 2) < 5 * 10^6 < T(5, 2) shows.
%e A367377 The corner of the table begins:
%e A367377   --------------------------------------------------------------------
%e A367377      Pattern   |   once    twice  3 times   4 times  5 times  6 times
%e A367377   --------------------------------------------------------------------
%e A367377         1      |      1        3        9        21       81      147
%e A367377        121     |      6       78     1014     12246   171366  1922622
%e A367377       12321    |     72    10728  1598472 230297976
%e A367377      1234321   |    120    28920  6969720
%e A367377     123454321  |   5184 53752896
%e A367377    12345654321 |   1440  4157280
%e A367377   1234567654321| 373248
%e A367377   ...
%e A367377 T(3, 4) must have 12 odd divisors and as least number must have 2^3 * 3^2 as a factor in order to create the initial width pattern 1 2 3 2 1 0. Therefore, since the next smallest prime larger than 16 * 9 is 149, T(3, 4) is 2^3 * 3^2 * 149^3 or 2^3 * 3^2 * 149 * p for suitable prime p which leads to p = 21467 < 22201 = 149^2.
%e A367377 All other numbers in the table were found by exhaustive computations.
%t A367377 t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]] (* row n in triangle of A249223 *)
%t A367377 t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)
%t A367377 widthPattern[n_] := Map[First, Split[t262045[n]]]
%t A367377 umw[n_, k_] := Most[Flatten[Table[Join[Range[n], Range[n-1, 0, -1]], k]]]
%t A367377 a367377[{n_, k_}, b_] := NestWhile[#+1&, 1, #<b&&widthPattern[#]!=umw[n, k]&] (* computation of T(n, k) stops at bound b *)
%Y A367377 Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596.
%K A367377 nonn,tabl,more
%O A367377 1,2
%A A367377 _Hartmut F. W. Hoft_, Nov 15 2023