A320521 -id:A320521 - cp's OEIS frontend

A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.

1, 3, 9, 21, 63, 147, 357, 903, 2499, 6069, 13915, 29095, 59455, 142945, 320045, 643885, 1367465, 3287735, 6779135, 13853015, 30262595, 61773745

Offset: 1

Omar E. Pol, Mar 23 2014

Conjecture 1: where records occur in A237271. - Omar E. Pol, Dec 27 2016
For more information about the symmetric representation of sigma see A237270, A237593.
This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - Hartmut F. W. Hoft, Dec 09 2014
Conjecture 2: all terms are odd numbers. - Omar E. Pol, Oct 14 2018
Proof of Conjecture 2: Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - Hartmut F. W. Hoft, Sep 22 2021
Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021
From Omar E. Pol, Jul 28 2025: (Start)
Conjecture 3: a(n) is the smallest number k having n 2-dense sublists of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
An example of the conjecture 3 for n = 1..5 is as shown below:
  ----------------------------------------------------
  |    |  List of divisors of k         |     |      |
  |  k |  [with sublists in brackets]   |  n  | a(n) |
  ----------------------------------------------------
  |  1 |  [1];                          |  1  |   1  |
  |  3 |  [1], [3];                     |  2  |   3  |
  |  9 |  [1], [3], [9];                |  3  |   9  |
  | 21 |  [1], [3], [7], [21];          |  4  |  21  |
  | 63 |  [1], [3], [7, 9], [21], [63]; |  5  |  63  |
(End)
Conjecture 4: a(n) is the smallest number k having n divisors p of k such that p is greater than twice the adjacent previous divisor of k. - Omar E. Pol, Aug 05 2025

			------------------------------------------------------
n       a(n)     A239665                  A266094(n)
------------------------------------------------------
1        1       [1]                           1
2        3       [2, 2]                        4
3        9       [5, 3, 5]                    13
4       21       [11, 5, 5, 11]               32
5       63       [32, 12, 16, 12, 32]        104
...
For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Hartmut F. W. Hoft, Procedural implementation for extension values
Omar E. Pol, Perspective view of the stepped pyramid (16 levels)

Row 1 of A240062.
Cf. A000203, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A238443, A239657, A239660, A239665, A239931-A239934, A245092, A262626, A266094.
Cf. A249223, A250070, A262045, A320521, A341969, A341970, A341971, A347980.
Cf. A174973 (2-dense numbers), A384149, A384222, A384225, A384226.

(* a239663[] permits computation in intervals *)
(* Function a237270[] is defined in A237270 *)
(* variable "list" contains the first occurrences up to m *)
a239663[list_,{m_, n_}]:=Module[{firsts=list, g=Length[list], i, p}, For[i=m, i<=n, i++, p=Length[a237270[i]]; If[p>g, AppendTo[firsts, i]; g=p]]; firsts]
a239663[{1}, {1, 1000}] (* computes the first 8 values *)
(* Hartmut F. W. Hoft, Jul 08 2014 *)
(* support functions are defined in A341969, A341970 & A341971 *)
a239663[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=1, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
a239663[62000000,22] (* Hartmut F. W. Hoft, Sep 22 2021 *)

a(6)-a(8) from Michel Marcus, Mar 28 2014
a(9) from Michel Marcus, Mar 29 2014
a(10)-a(11) from Michel Marcus, Apr 02 2014
a(12) from Hartmut F. W. Hoft, Jul 08 2014
a(13)-a(18) from Hartmut F. W. Hoft, Dec 09 2014
a(19)-a(22) from Hartmut F. W. Hoft, Sep 22 2021

A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

Original entry on oeis.org

2, 4, 10, 6, 14, 50, 8, 22, 70, 230, 12, 26, 98, 250, 1150, 16, 34, 110, 290, 1250, 5050, 18, 38, 130, 310, 1450, 5150, 22310, 20, 44, 154, 370, 1550, 5290, 23230, 106030, 24, 46, 170, 406, 1850, 5350, 23690, 106490, 510050, 28, 52, 182, 410, 2030, 5450, 24610, 107410, 513130, 2065450

Offset: 1

Omar E. Pol, Oct 15 2018

This is a permutation of the positive even numbers (A299174).
The union of all odd-indexed columns gives A319796, the even numbers in A071562.
The union of all even-indexed columns gives A319802, the even numbers in A071561.

			From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
The 10x10 section of table T(n,k):
(Table with first 20 terms from _Omar E. Pol_)
------------------------------------------------------------------
n\k | 1   2   3    4    5     6     7      8       9       10  ...
------------------------------------------------------------------
  1 | 2   10  50   230  1150  5050  22310  106030  510050  2065450
  2 | 4   14  70   250  1250  5150  23230  106490  513130  2115950
  3 | 6   22  98   290  1450  5290  23690  107410  520150  2126050
  4 | 8   26  110  310  1550  5350  24610  110170  530150  2157850
  5 | 12  34  130  370  1850  5450  25070  112010  530450  2164070
  6 | 16  38  154  406  2030  5650  25250  112930  532450  2168150
  7 | 18  44  170  410  2050  5750  25750  114770  534290  2176550
  8 | 20  46  182  430  2150  6250  25990  115690  537050  2186650
  9 | 24  52  190  434  2170  6350  26450  116150  540350  2216950
  10| 28  58  238  470  2350  6550  26750  117070  544870  2219650
   ... (End)

Row 1 is A320521.
Column 1 gives A174973 = A238443, without the 1.
Column 2 gives A244894.
Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]Hartmut F. W. Hoft, Oct 06 2021 *)

Terms a(21) and beyond from Hartmut F. W. Hoft, Oct 06 2021

cp's OEIS Frontend

A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.

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