cp's OEIS Frontend

A342595 Irregular triangle of A342592 read by rows arranged first by length of the width pattern and then lexicographically within blocks of patterns of equal length.

%I A342595 #10 Mar 18 2021 23:52:30
%S A342595 1,1,0,1,1,2,1,1,0,1,0,1,1,2,1,2,1,1,2,3,2,1,1,0,1,0,1,0,1,1,0,1,2,1,
%T A342595 0,1,1,2,1,0,1,2,1,1,2,1,2,1,2,1,1,2,3,2,3,2,1,1,2,3,4,3,2,1,1,0,1,0,
%U A342595 1,0,1,0,1,1,2,1,2,1,2,1,2,1,1,2,3,2,3,2,3,2,1
%N A342595 Irregular triangle of A342592 read by rows arranged first by length of the width pattern and then lexicographically within blocks of patterns of equal length.
%C A342595 Let u and v be two symmetric width patterns; then u < v if u is shorter than v, and if they have the same length then they are ordered lexicographically, i.e., if i is the first index where u and v differ and u(i) < v(i) then u < v.
%C A342595 This sequence is a permutation of the rows of the irregular triangle in A342592. Every row in the triangle representing a width pattern w contains an odd number 2*k - 1, k >= 1, of entries where k is the number of odd divisors of the smallest number whose symmetric representation of sigma realizes that pattern.
%C A342595 The number of distinct width patterns w of the same length 2*k-1 created by numbers with k odd divisors is computationally challenging since the numbers of their first occurrence can be very large (A342592, A342596). The counts in the table below are already established for n <= 5*10^5 and have not changed through 10^7; the counts are not stable at that larger level for width patterns of numbers with more than 8 odd divisors:
%C A342595 # odd divisors   1   2   3   4    5    6    7    8
%C A342595 pattern count    1   2   3   6    5   16    7   40
%C A342595 A001405          1   2   3   6   10   20   35   70
%C A342595 For any odd number q with k divisors and 2^s < q < 2^(s+1), s >= 0, any number 2^t * q with t > s has the lexicographically largest symmetric width pattern 1 2 3 ... k-2 k-1 k k-1 k-2 ... 3 2 1 of length 2*k - 1. Therefore, the sequence q, 2 * q, 2^2 * q, ... , 2^s * q instantiates at most s+1 different symmetric width patterns; these range from 2 for prime numbers q, patterns (1 0 1) and (1 2 1), to the maximum of s+1 different patterns such as for q = 105 = 3*5*7.
%e A342595 The number of entries through the center in a row of the triangle below equals the number of odd divisors of any number that has that pattern of widths.
%e A342595 The pattern in row 10 of the triangle below, realized first by n = 30 which labels the row is the smallest number with width pattern (1 2 1 2 1 2 1); 42, 54 and 66 are the other numbers less than 100 realizing that pattern.
%e A342595 The triangle below lists the first 21 distinct symmetric width patterns in the order described above. The smallest number whose symmetric representation of sigma has the width pattern of that row is listed as first column (see A342596). All possible symmetric width patterns of lengths 1, 3, 5 and 7 are realized in the triangle below; their respective counts are A001405(1,2,3,4) = (1,2,3,6).
%e A342595      1   1
%e A342595      3   1 0 1
%e A342595      6   1 2 1
%e A342595      9   1 0 1 0 1
%e A342595     18   1 2 1 2 1
%e A342595     72   1 2 3 2 1
%e A342595     21   1 0 1 0 1 0 1
%e A342595     15   1 0 1 2 1 0 1
%e A342595     78   1 2 1 0 1 2 1
%e A342595     30   1 2 1 2 1 2 1
%e A342595     60   1 2 3 2 3 2 1
%e A342595    120   1 2 3 4 3 2 1
%e A342595     81   1 0 1 0 1 0 1 0 1
%e A342595    162   1 2 1 2 1 2 1 2 1
%e A342595    648   1 2 3 2 3 2 3 2 1
%e A342595   1296   1 2 3 4 3 4 3 2 1
%e A342595   5184   1 2 3 4 5 4 3 2 1
%e A342595    147   1 0 1 0 1 0 1 0 1 0 1
%e A342595     63   1 0 1 0 1 2 1 0 1 0 1
%e A342595     75   1 0 1 2 1 0 1 2 1 0 1
%e A342595     45   1 0 1 2 1 2 1 2 1 0 1
%t A342595 (* function a341969[ ] is defined in A341969 *)
%t A342595 lexicographic[s1_, s2_] := Module[{k=1}, While[s1[[2, k]]==s2[[2, k]], k++]; s1[[2,k]]<s2[[2,k]]]
%t A342595 lexicographicOrder[s1_, s2_] := Module[{s1L=Length[s1[[2]]], s2L=Length[s2[[2]]]},( s1L<s2L)||(s1L==s2L&&lexicographic[s1, s2])]
%t A342595 a342595[n_] := Module[{listW={}, listK={}, k, w}, For[k=1,k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; Map[Last, Sort[Transpose[{listK, listW}], lexicographicOrder]]]
%t A342595 Flatten[Take[a342595[1000], 15]]  (* entries of the first 15 rows of the triangle *)
%Y A342595 Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342596.
%K A342595 nonn,tabf
%O A342595 1,6
%A A342595 _Hartmut F. W. Hoft_, Mar 16 2021