A347980 -id:A347980 - 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

A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

Original entry on oeis.org

1, 2, 6, 3, 12, 60, 4, 15, 72, 120, 5, 18, 84, 180, 360, 7, 20, 90, 240, 420, 840, 8, 24, 126, 252, 720, 1080, 3360, 9, 28, 140, 336, 1008, 1260, 3600, 2520, 10, 30, 144, 378, 1200, 1440, 3780, 5544, 5040, 11, 35, 168, 432, 1320, 1680, 3960, 6300, 7560, 10080, 13, 36, 198, 480, 1512, 1800, 4200, 6720, 9240, 12600, 15120

Offset: 1

Omar E. Pol, Jul 08 2015

This is a permutation of the natural numbers.
Row 1 gives A250070.
For more information about the widths of the symmetric representation of sigma see A249351 and A250068.
The next term: 120 < T(2,4) < 360.
From Hartmut F. W. Hoft, Sep 20 2024: (Start)
Column T(j,1), j>=1, forms A174905 and is a permutation of A357581. Numbers T(j,k), j>=1 and k>1, form A005279. Conjecture: Every column of the square array contains odd numbers.
The sequence of smallest odd numbers in each column forms A347980. E.g., in column 12 the smallest odd number is T(466, 12) = 765765 = A347980(12) which is equivalent to A250068(765765) = 12. (End)

			The corner of the square array T(j,k) begins:
  1,  6, 60, 120, 360, ...
  2, 12, 72, ...
  3, 15, 84, ...
  4, 18, ...
  5, 20, ...
  7, ...
  ...
For j = 1 and k = 2; T(1,2) is the first number n such that the symmetric representation of sigma(n) has a part with maximum width 2 as shown below:
.
      Dyck paths            Cells              Widths
      _ _ _ _             _ _ _ _
      _ _ _  |_          |_|_|_|_|_          / / / /
           |   |_              |_|_|_              / /
           |_ _  |             |_|_|_|             / / /
               | |                 |_|                 /
               | |                 |_|                 /
               | |                 |_|                 /
.
The widths of the symmetric representation of sigma(6) = 12 are [1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1], also the 6th row of triangle A249351.
From _Hartmut F. W. Hoft_, Sep 20 2024: (Start)
Extending the terms T(j,k) to a 12x12 square array:
j\k 1  2  3   4   5    6    7    8     9     10    11    12
--------------------------------------------------------------
1 | 1  6  60  120 360  840  3360 2520  5040  10080 15120 32760
2 | 2  12 72  180 420  1080 3600 5544  7560  12600 20160 36960
3 | 3  15 84  240 720  1260 3780 6300  9240  13860 25200 39600
4 | 4  18 90  252 1008 1440 3960 6720  10920 15840 35280 41580
5 | 5  20 126 336 1200 1680 4200 6930  11880 16380 40320 43680
6 | 7  24 140 378 1320 1800 4320 7140  14040 16800 42840 45360
7 | 8  28 144 432 1512 1980 4620 7920  16632 18480 46800 46200
8 | 9  30 168 480 1560 2016 4680 8190  17160 18900 47880 47520
9 | 10 35 198 504 1848 2100 5280 8400  17640 21420 56160 49140
10| 11 36 210 540 1890 2160 5400 9360  18720 21840 56700 51480
11| 13 40 216 594 2184 2340 5460 10296 19800 22680 57120 52920
12| 14 42 264 600 2310 2640 5940 10800 20790 23760 57960 54600
...
(End)

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..820
Index entries for sequences that are permutations of the natural numbers

Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A250071.

Cf. A005279, A174905, A347980, A357581.

(* Computing table T(j,k) of size mxn with bound b *)
eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mWidth[n_] :=Max[FoldList[#1+If[OddQ[#2], 1, -1]&, sDiv[n]]]
t253258[{m_, n_}, b_] := Module[{s=Table[0, {i, m+1}, {j, n}], k=1, w, f}, While[k<=b, w=mWidth[k]; If[w<=n, f=s[[m+1, w]]; If[fHartmut F. W. Hoft, Sep 20 2024 *)

More terms from Charlie Neder, Jan 11 2019

A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

Original entry on oeis.org

2, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440

Offset: 1

Hartmut F. W. Hoft, Sep 22 2021

For the 30 known terms the symmetric representation of sigma consists of a single part, i.e., this is a subsequence of A174973 = A238443.
The sequence is not increasing with the maximum width of the symmetric representation of sigma.
Also a(33) = 2162160 is the only further number in the sequence less than 2500000.

			The pattern of maximum widths within the single part of the symmetric representation of sigma for the first four numbers in the sequence is:
  a(n) parts successive widths
    2:   1           1
    6:   1         1 2 1
   60:   1     1 2 3 2 3 2 1
  120:   1     1 2 3 4 3 2 1

Cf. A174973, A237048, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A250070, A262045, A341969, A341970, A341971, A347980.

a262045[n_] := Module[{a=Accumulate[Map[If[Mod[n - # (#+1)/2, #]==0, (-1)^(#+1), 0] &, Range[Floor[(Sqrt[8n+1]-1)/2]]]]}, Join[a, Reverse[a]]]
a347979[n_, mw_] := Module[{list=Table[0, mw], i, v}, For[i=2, i<=n, i+=2, v=Max[a262045[i]]; If [list[[v]]==0, list[[v]]=i]]; list]
a347979[2500000, 33] (* computes a(1..30), a(33); a(31..32) > 2500000 *)

It appears that a(n) = A250070(n) if n >= 2.

A348171 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in at least one of its p parts.

Original entry on oeis.org

1, 6, 3, 60, 78, 9, 120, 7620, 15, 21, 360, 28920, 315, 75, 81, 840, 261720, 1326, 495, 63, 147, 3360, 1422120, 3465, 22542, 525, 189, 729, 2520, 22622880, 17325, 44574, 5005, 1275, 357, 903, 5040, 12728520, 45045, 199578, 6435, 16575, 1287, 1197, 3025, 10080, 50858640, 51975, 7734558, 34034, 131835, 2145, 3861, 2499, 6875

Offset: 1

Hartmut F. W. Hoft, Oct 04 2021

The first row of the table below is A318843 and the first column is A250070.
T(1,k+1) <= 3^k, for all k>=0, since for k=2j the (j+1)-st part in the symmetric representation of sigma(3^k) extends across the diagonal, and for k=2j+1 the (j+1)-st part is completed before the diagonal.
The data computed so far for a partially filled table of 15 rows and 15 columns, show that all rows, all columns (except column 4 for n <= 6 *10^7), and the diagonal are nonmonotonic.

			The 10x10 section of the table with dashes indicating values greater than 6*10^7; rows w denote the maximum width and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
w\p | 1     2        3      4       5       6       7       8        9   ...
----------------------------------------------------------------------------
  1 | 1     3        9      21      81      147     729     903      3025
  2 | 6     78       15     75      63      189     357     1197     2499
  3 | 60    7620     315    495     525     1275    1287    3861     3591
  4 | 120   28920    1326   22542   5005    16575   2145    29325    11583
  5 | 360   261720   3465   44574   6435    131835  76125   24225    82593
  6 | 840   1422120  17325  199578  34034   83655   196707  468027   62985
  7 | 3360  22622880 45045  7734558 153153  442442  314925  1108965  471975
  8 | 2520  12728520 51975     -    205275  2067065 1429275 2359875  557175
  9 | 5040  50858640 225225    -    646646  2863718 2395197 5353725  2785875
  10| 10080    -     405405    -    1990989 2124694 6500375 36535499 7753875
   ...
The symmetric representation of sigma for T(2,3) = 15 consists of the three parts (8, 8, 8) of maximum widths (1, 2, 1), and that of T(3,3) = 315 consists of the three parts (158, 308, 158) of maximum widths (1, 3, 1).

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348142.

(* function a341969 is defined in A341969 *)
a348171[n_,  {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], # != 0 &], #[[1]] != 0 &]]; c = Length[s]; u = Max[s]; If[u<=w && c<=p, If[list[[u, c]] == 0, list[[u, c]] = k ]]]; list]
table=a348171[60000000, {15, 15}] (* 15x15 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&, Range[55]] (* sequence data *)

a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.

T(w(n), p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2, w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).

A348142 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in everyone of its p parts.

Original entry on oeis.org

1, 6, 3, 60, 78, 9, 120, 7620, 1014, 21, 360, 28920, 967740, 12246, 81, 840, 261720, 6969720, 116136420, 171366, 147, 3360, 1422120

Offset: 1

Hartmut F. W. Hoft, Oct 04 2021

It appears that the first row is A318843 and that the first column is A250070.
Columns 1 and 2 both are identical with those of the table in A348171 and row 1 is identical with that of A348171.
In the remainder of the 7th antidiagonal a(24..26) > 120*10^6, a(27) = 1922622, and a(28) = 903.

			The 10x8 section of the table T(w,p) with dashes indicating values greater than 120*10^6; rows w denote the common maximum width in all parts and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
w\p | 1     2         3        4         5        6        7        8  ...
--------------------------------------------------------------------------
  1 | 1     3         9        21        81       147     729      903
  2 | 6     78        1014     12246     171366   1922622 28960854  -
  3 | 60    7620      967740   116136420   -         -       -
  4 | 120   28920     6969720    -
  5 | 360   261720      -
  6 | 840   1422120     -
  7 | 3360  22622880    -
  8 | 2520  12728520    -
  9 | 5040  50858640    -
  10| 10080    -
   ...
The symmetric representation of sigma for T(2,2) = 78 consists of the two parts (84, 84) of maximum widths (2, 2), and that of T(2,3) = 1014 consists of the three parts (1020, 156, 1020) of maximum widths (2, 2, 2).

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348171.

(* function a341969 is defined in A341969 *)
a348142[n_, {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, Monitor[For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], #!=0&], #[[1]]!=0&]]; c=Length[s]; u=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]], list]; list]
table=a348142[120000000, {10, 10}] (* 10x10 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&,  Range[23]] (* sequence data *)

a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.

T(w(n),p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2, w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).

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

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A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

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A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

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A348171 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in at least one of its p parts.

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A348142 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in everyone of its p parts.

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