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

Author

Omar E. Pol, Mar 23 2014

Keywords

• nonn
• more
• hard

Comments

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

Examples

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

Links

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

Crossrefs

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.

Programs

Mathematica

(* 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 *)

Extensions

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