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%I #10 Nov 23 2020 07:56:20
%S 6,15,28,91,153,190,325,496,703,861,946,1431,1891,2278,2701,3655,4753,
%T 5151,5356,5995,6441,7381,8128,8911,9453,10585,11476,12403,13366,
%U 15051,18721,21115,22366,23653
%N Numbers n whose symmetric representation of sigma(n) has a maximum width of 2 that occurs exactly once (at the diagonal).
%C For numbers n in the sequence the symmetric representation of sigma(n) consists of an odd number of regions. The geometric form for a single width 2 at the diagonal requires that
%C (a) the length of the last leg of the n-th Dyck path is a237591(n,row(n)) = 1 and the leg horizontal, which implies that row(n) is odd, and
%C (b) the length of the last leg of the (n-1)-st Dyck path is a237591(n-1,row(n-1)) = 2 and the leg vertical, which implies that row(n-1) = row(n) - 1 is even:
%C |_
%C | |_
%C |_ _
%C Therefore, n is a hexagonal number (see A000384) and row(n) is an odd divisor of n since otherwise the central region of the symmetric representation for sigma(n) defined by the largest odd divisor d < row(n) would create an extent of width 2 larger than 1.
%C Since all other regions must have width 1 the following conditions characterize the numbers in this sequence:
%C Let 1 = d_1 < d_2 < ... < d_s <= row(n) < d_(s+1) < ... < d_t = q be all odd divisors of n = 2^k * q, k >= 0. n is hexagonal, 2^(k+1) * d_i < d_(i+1), for 1 <= i <= s-2, and d_s = row(n), so that 2^(k+1) * d_(s-1) = d_s + 1.
%C The numbers of this sequence can be arranged as a table according to the number of regions (one fewer than the number of its odd divisors) in the symmetric representation of sigma(n):
%C 1 3 5 7 9 11
%C -------------------------------------------------
%C 6 15 153 861 195625 43071
%C 28 91 325 1431 ... ...
%C 496 190 4753 3655 6859425628 50999950
%C 8128 703 7381 5151 ... ...
%C ... 946 ... 5995
%C 1891 468028 6441
%C 2278 ... 8911
%C 2701 9453
%C 5356 10585
%C 11476 15051
%C 12403 21115
%C 13366 23653
%C 18721 ...
%C 22366 124750
%C ... ...
%C The numbers less than 25000 in the first four columns were computed using function a338488[] while the numbers in the remaining two columns and the first even numbers in the 5-column and 7-column were computed by a function implementing the conditions on the structure of the odd divisors.
%C The 1-column consists of the even perfect numbers, A000396.
%C The 3-column is the sequence of numbers n =2^k * p * q, p & q odd primes, such that 2^(k+1) < p < q < 2^(k+1)*p = q+1. It is a subsequence of A338486, and includes the odd numbers in A129521 since (q+1)/2 = p is prime.
%C The odd numbers in the 5-column have 6 divisors and therefore form a subsequence in A116565.
%C Conjecture: For every odd number 2k-1 there is an even number n in this sequence whose symmetric representation of sigma(n) has 2k-1 regions.
%e a(5) = 153 = 17*3^2 is in the sequence and in the 5-column of the table since 1 < 2 < 3 < 6 < 3^2 < 17 = row(153) < 2*3^2 representing the 6 odd divisors 1 - 153 - 3 - 51 - 9 - 17 (see A237048) results in the following pattern for the widths of its 17 legs (see A249223): 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 5 regions with a single unit of width 2.
%e a(6) = 190 = 2*5*19 is in the sequence and in the 3-column of the table since 1 < 4 < 5 < 19 = row(190) < 4*5 representing the 4 odd divisors 1 - 190 - 5 - 19 results in the following pattern for the widths of its 19 legs: 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with a single unit of width 2.
%e Number 66 = 2*3*11 is not in the sequence since positions 1 < 3 < 4 < 11 = row(66) < 4*3 representing the 4 odd divisors 1 - 3 - 33 - 11 violate the condition 4 = 4*d_1 < d_2 = 3; its symmetric representation of sigma consists of a single region in which the third leg and its symmetric copy have width 2 in addition to a single unit of width 2 at the diagonal.
%t (* function path[] and support functions are defined in A237270 *)
%t a338488[m_, n_] := Module[{p0=path[m-1], p1, k, srs, w2, list={}}, For[k=m, k<=n, k++, p1=path[k]; srs=Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[p1, 1], -1], p0}]]; w2=Length[Select[srs, #=={2, 2}&]]; If[Max[srs]==2&&w2==1, AppendTo[list, k]]; p0=p1]; list]
%t a338488[1,25000] (* sequence data *)
%Y Cf. A000384, A000396, A116565, A129521, A237048, A237270, A237591, A237593, A249223, A338488.
%K nonn
%O 1,1
%A _Hartmut F. W. Hoft_, Oct 30 2020
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