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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Mar 23 2014

Keywords

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

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

A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

Original entry on oeis.org

1, 2, 2, 5, 3, 5, 11, 5, 5, 11, 32, 12, 16, 12, 32, 74, 26, 14, 14, 26, 74, 179, 61, 29, 38, 29, 61, 179, 452, 152, 68, 32, 32, 68, 152, 452, 1250, 418, 182, 152, 100, 152, 182, 418, 1250, 3035, 1013, 437, 342, 85, 85, 342, 437, 1013, 3035, 6958, 1394, 638, 314, 154, 236, 154, 314, 638, 1394, 6958
Offset: 1

Views

Author

Omar E. Pol, Mar 23 2014

Keywords

Comments

Row n is also row A239663(n) of A237270.

Examples

			----------------------------------------------------------------------
n    A239663(n)  Triangle begins:                        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
6      147       [74, 26, 14, 14, 26, 74]                  228
7      357       [179, 61, 29, 38, 29, 61, 179]            576
8      903       [452, 152, 68, 32, 32, 68, 152, 452]     1408
...
Illustration of initial terms:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.     _ _ 2          | |
.    |_ _|_ 2        | |
.     _ 1| |         | |
.    |_| |_|         |_|
.
For n = 2 we have that A239663(2) = 3 is the smallest number whose symmetric representation of sigma has 2 parts. Row 3 of A237593 is [2, 1, 1, 2] and row 2 of A237593 is [2, 2] therefore between both Dyck paths in the first quadrant there are two regions (or parts) of sizes [2, 2], so row 2 is [2, 2].
For n = 3 we have that A239663(3) = 9 is the smallest number whose symmetric representation of sigma has 3 parts. The 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both Dyck paths in the first quadrant there are three regions (or parts) of sizes [5, 3, 5], so row 3 is [5, 3, 5].

Crossrefs

Cf. A000203, A005279, A196020, A236104, A237270, A237271, A235791, A237591, A237593, A239660, A239663, A239931-A239934, A240020, A240062, A244050, A245092, A262626, A266094.

Extensions

a(16)-a(28) from Michel Marcus and Omar E. Pol, Mar 28 2014
a(29)-a(36) from Michel Marcus, Mar 28 2014
a(37)-a(45) from Michel Marcus, Mar 29 2014
a(46)-a(66) from Michel Marcus, Apr 02 2014

A357775 Numbers k with the property that the symmetric representation of sigma(k) has seven parts.

Original entry on oeis.org

357, 399, 441, 483, 513, 567, 609, 621, 651, 729, 759, 777, 783, 837, 861, 891, 957, 999, 1023, 1053, 1089, 1107, 1131, 1161, 1209, 1221, 1269, 1287, 1323, 1353, 1419, 1431, 1443, 1521, 1551, 1595, 1599, 1677, 1705, 1749, 1815, 1833, 1887, 1947, 1989, 2013, 2035, 2067, 2091, 2145, 2193, 2223, 2255
Offset: 1

Views

Author

Omar E. Pol, Oct 12 2022

Keywords

Examples

			357 is in the sequence because the 357th row of A237593 is [179, 60, 31, 18, 12, 9, 7, 6, 4, 4, 3, 3, 2, 3, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 4, 4, 6, 7, 9, 12, 18, 31, 60, 179], and the 356th row of the same triangle is [179, 60, 30, 18, 13, 9, 6, 6, 4, 4, 3, 3, 3, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 3, 3, 3, 4, 4, 6, 6, 9, 13, 18, 30, 60, 179], therefore between both symmetric Dyck paths there are seven parts: [179, 61, 29, 38, 29, 61, 179].
Note that the sum of these parts is 179 + 61 + 29 + 38 + 29 + 61 + 179 = 576, equaling the sum of the divisors of 357: 1 + 3 + 7 + 17 + 21 + 51 + 119 + 357 = 576.
(The diagram of the symmetric representation of sigma(357) = 576 is too large to include.)

Crossrefs

Column 7 of A240062.
Cf. A237270 (the parts), A237271 (number of parts), A238443 = A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts), A320511 (six parts).
Cf. A018411, A196020, A236104, A235791, A237591, A237593, A239663, A266094.

Formula

A237271(a(n)) = 7.
Showing 1-3 of 3 results.

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