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A338488
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Numbers n whose symmetric representation of sigma(n) has a maximum width of 2 that occurs exactly once (at the diagonal).
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2
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6, 15, 28, 91, 153, 190, 325, 496, 703, 861, 946, 1431, 1891, 2278, 2701, 3655, 4753, 5151, 5356, 5995, 6441, 7381, 8128, 8911, 9453, 10585, 11476, 12403, 13366, 15051, 18721, 21115, 22366, 23653
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OFFSET
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1,1
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COMMENTS
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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
(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
(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:
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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.
Since all other regions must have width 1 the following conditions characterize the numbers in this sequence:
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.
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):
1 3 5 7 9 11
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6 15 153 861 195625 43071
28 91 325 1431 ... ...
496 190 4753 3655 6859425628 50999950
8128 703 7381 5151 ... ...
... 946 ... 5995
1891 468028 6441
2278 ... 8911
2701 9453
5356 10585
11476 15051
12403 21115
13366 23653
18721 ...
22366 124750
... ...
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.
The 1-column consists of the even perfect numbers, A000396.
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.
The odd numbers in the 5-column have 6 divisors and therefore form a subsequence in A116565.
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.
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LINKS
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EXAMPLE
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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.
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.
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.
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MATHEMATICA
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(* function path[] and support functions are defined in A237270 *)
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]
a338488[1, 25000] (* sequence data *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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