A338486 -id:A338486 - cp's OEIS frontend

A338488 Numbers n whose symmetric representation of sigma(n) has a maximum width of 2 that occurs exactly once (at the diagonal).

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

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

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:
    |_
  |   |_
  | 
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
  -------------------------------------------------
     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.

			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.

Cf. A000384, A000396, A116565, A129521, A237048, A237270, A237591, A237593, A249223, A338488.

(* 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 *)

cp's OEIS Frontend

A338488 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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