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

%I A239663 #101 Aug 11 2025 23:50:46
%S A239663 1,3,9,21,63,147,357,903,2499,6069,13915,29095,59455,142945,320045,
%T A239663 643885,1367465,3287735,6779135,13853015,30262595,61773745
%N A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.
%C A239663 Conjecture 1: where records occur in A237271. - _Omar E. Pol_, Dec 27 2016
%C A239663 For more information about the symmetric representation of sigma see A237270, A237593.
%C A239663 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
%C A239663 Conjecture 2: all terms are odd numbers. - _Omar E. Pol_, Oct 14 2018
%C A239663 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
%C A239663 Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - _Omar E. Pol_, Sep 22 2021
%C A239663 From _Omar E. Pol_, Jul 28 2025: (Start)
%C A239663 Conjecture 3: a(n) is the smallest number k having n 2-dense sublists of divisors of k.
%C A239663 The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
%C A239663 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.
%C A239663 An example of the conjecture 3 for n = 1..5 is as shown below:
%C A239663   ----------------------------------------------------
%C A239663   |    |  List of divisors of k         |     |      |
%C A239663   |  k |  [with sublists in brackets]   |  n  | a(n) |
%C A239663   ----------------------------------------------------
%C A239663   |  1 |  [1];                          |  1  |   1  |
%C A239663   |  3 |  [1], [3];                     |  2  |   3  |
%C A239663   |  9 |  [1], [3], [9];                |  3  |   9  |
%C A239663   | 21 |  [1], [3], [7], [21];          |  4  |  21  |
%C A239663   | 63 |  [1], [3], [7, 9], [21], [63]; |  5  |  63  |
%C A239663 (End)
%C A239663 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
%H A239663 Hartmut F. W. Hoft, <a href="/A239663/a239663.pdf">Procedural implementation for extension values</a>
%H A239663 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (16 levels)</a>
%e A239663 ------------------------------------------------------
%e A239663 n       a(n)     A239665                  A266094(n)
%e A239663 ------------------------------------------------------
%e A239663 1        1       [1]                           1
%e A239663 2        3       [2, 2]                        4
%e A239663 3        9       [5, 3, 5]                    13
%e A239663 4       21       [11, 5, 5, 11]               32
%e A239663 5       63       [32, 12, 16, 12, 32]        104
%e A239663 ...
%e A239663 For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
%e A239663 .
%e A239663 .     _ _ _ _ _ 5
%e A239663 .    |_ _ _ _ _|
%e A239663 .              |_ _ 3
%e A239663 .              |_  |
%e A239663 .                |_|_ _ 5
%e A239663 .                    | |
%e A239663 .                    | |
%e A239663 .                    | |
%e A239663 .                    | |
%e A239663 .                    |_|
%e A239663 .
%t A239663 (* a239663[] permits computation in intervals *)
%t A239663 (* Function a237270[] is defined in A237270 *)
%t A239663 (* variable "list" contains the first occurrences up to m *)
%t A239663 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]
%t A239663 a239663[{1}, {1, 1000}] (* computes the first 8 values *)
%t A239663 (* _Hartmut F. W. Hoft_, Jul 08 2014 *)
%t A239663 (* support functions are defined in A341969, A341970 & A341971 *)
%t A239663 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]
%t A239663 a239663[62000000,22] (* _Hartmut F. W. Hoft_, Sep 22 2021 *)
%Y A239663 Row 1 of A240062.
%Y A239663 Cf. A000203, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A238443, A239657, A239660, A239665, A239931-A239934, A245092, A262626, A266094.
%Y A239663 Cf. A249223, A250070, A262045, A320521, A341969, A341970, A341971, A347980.
%Y A239663 Cf. A174973 (2-dense numbers), A384149, A384222, A384225, A384226.
%K A239663 nonn,more,hard
%O A239663 1,2
%A A239663 _Omar E. Pol_, Mar 23 2014
%E A239663 a(6)-a(8) from _Michel Marcus_, Mar 28 2014
%E A239663 a(9) from _Michel Marcus_, Mar 29 2014
%E A239663 a(10)-a(11) from _Michel Marcus_, Apr 02 2014
%E A239663 a(12) from _Hartmut F. W. Hoft_, Jul 08 2014
%E A239663 a(13)-a(18) from _Hartmut F. W. Hoft_, Dec 09 2014
%E A239663 a(19)-a(22) from _Hartmut F. W. Hoft_, Sep 22 2021