A341971 -id:A341971 - cp's OEIS frontend

A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

2, 10, 50, 230, 1150, 5050, 22310, 106030, 510050, 2065450, 10236350

Offset: 1

Omar E. Pol, Oct 14 2018

It appears that a(n) = 2 * q where q is odd and that the symmetric representation of sigma(a(n)/2) has the same number of parts as that for a(n). Number a(12) > 15000000. - Hartmut F. W. Hoft, Sep 22 2021

			a(1) = 2 because the second row of A237593 is [2, 2], and the first row of the same triangle is [1, 1], therefore between both symmetric Dyck paths there is only one part: [3], equaling the sum of the divisors of 2: 1 + 2 = 3. See below:
.
.     _ _ 3
.    |_  |
.      |_|
.
.
a(2) = 10 because the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6], and the 9th row of the same triangle is [5, 2, 2, 2, 2, 5], therefore between both symmetric Dyck paths there are two parts: [9, 9]. Also there are no even numbers k < 10 whose symmetric representation of sigma(k) has two parts. Note that the sum of these parts is 9 + 9 = 18, equaling the sum of the divisors of 10: 1 + 2 + 5 + 10 = 18. See below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.
a(3) = 50 because the 50th row of A237593 is [26, 9, 4, 3, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 3, 4, 9, 26], and the 49th row of the same triangle is [25, 9, 5, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 5, 9, 25], therefore between both symmetric Dyck paths there are three parts: [39, 15, 39]. Also there are no even numbers k < 50 whose symmetric representation of sigma(k) has three parts. Note that the sum of these parts is 39 + 15 + 39 = 93, equaling the sum of the divisors of 50: 1 + 2 + 5 + 10 + 25 + 50 = 93. (The diagram of the symmetric representation of sigma(50) = 93 is too large to include.)

Row 1 of A320537.
Cf. A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts), A320511 (six parts).
Cf. A000203, A018262, A005843, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A240062, A245092, A262626, A296508.
Cf. A250070, A262045, A341969, A341970, A341971, A347979.

(* support functions are defined in A341969, A341970 & A341971 *)
a320521[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=2, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
a320521[15000000,11] (* Hartmut F. W. Hoft, Sep 22 2021 *)

a(6)-a(11) 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

A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 6, 1, 7, 3, 8, 9, 4, 2, 10, 1, 11, 5, 12, 3, 13, 6, 14, 2, 15, 7, 4, 1, 16, 17, 8, 18, 5, 3, 19, 9, 20, 2, 21, 10, 6, 1, 22, 4, 23, 11, 24, 7, 25, 12, 3, 26, 5, 27, 13, 8, 2, 28, 1, 29, 14, 30, 9, 6, 4, 31, 15, 32, 33, 16, 10, 3, 34, 7, 35, 17, 5, 2, 36, 11, 1, 37, 18, 38, 8, 39, 19, 12, 4, 40, 6, 41, 20

Offset: 1

Omar E. Pol, Dec 31 2024

Row n gives the first A001227(n) terms of the n-th row of A379630.

			Triangle begins:
   1;
   2;
   3,  1;
   4;
   5,  2;
   6,  1;
   7,  3;
   8;
   9,  4,  2;
  10,  1;
  11,  5;
  12,  3;
  13,  6;
  14,  2;
  15,  7,  4,  1;
  16;
  17,  8;
  18,  5,  3;
  19,  9;
  20,  2;
  21, 10,  6,  1;
  22,  4;
  23, 11;
  24,  7;
  25, 12,  3;
  26,  5;
  27, 13,  8,  2;
  28,  1;
  ...
Illustration of initial terms:
                                                         _
                                                       _|1|
                                                     _|2 _|
                                                   _|3  |1|
                                                 _|4   _| |
                                               _|5    |2 _|
                                             _|6     _| |1|
                                           _|7      |3  | |
                                         _|8       _|  _| |
                                       _|9        |4  |2 _|
                                     _|10        _|   | |1|
                                   _|11         |5   _| | |
                                 _|12          _|   |3  | |
                               _|13           |6    |  _| |
                             _|14            _|    _| |2 _|
                           _|15             |7    |4  | |1|
                         _|16              _|     |   | | |
                       _|17               |8     _|  _| | |
                     _|18                _|     |5  |3  | |
                   _|19                 |9      |   |  _| |
                 _|20                  _|      _|   | |2 _|
               _|21                   |10     |6   _| | |1|
             _|22                    _|       |   |4  | | |
           _|23                     |11      _|   |   | | |
         _|24                      _|       |7    |  _| | |
       _|25                       |12       |    _| |3  | |
     _|26                        _|        _|   |5  |  _| |
   _|27                         |13       |8    |   | |2 _|
  |28                           |         |     |   | | |1|
  ...
The diagram is also the left part of the diagram of A379630.
The geometrical structure is the same as the diagram of A237591.

Positive terms of A211343.
Absolute values of A341971.
Column 1 gives A000027.
Right border gives A118235.
Row lengths give A001227.
Row sums give A286014.
Subsequence of A286001 and of A299765 and of A379630.
For the largest parts see A379633.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379631, A379634.

A346969 1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 parts.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 21, 11, 25, 27, 63, 13, 35, 33, 81, 147, 17, 45, 39, 99, 171, 357, 19, 49, 51, 117, 189, 399, 903, 23, 77, 55, 153, 207, 441, 987, 2499, 29, 91, 57, 165, 243, 483, 1029, 2709, 6069, 31, 121, 65, 195, 261, 513, 1113, 2793, 6321, 13915, 37, 135, 69, 231, 275, 567, 1197, 2961, 6325, 14847, 29095

Offset: 1

Hartmut F. W. Hoft, Oct 06 2021

This sequence is a permutation of the odd positive integers.

The first row of table T(n,k) preceded by a(1) = 1 is A239663; the first column is the sequence A065091 of odd primes; the second column contains the squares of the odd primes as a subsequence (see also A247687).

			The 10x10 initial submatrix of table T(n,k):
n\k | 2   3    4    5     6     7     8      9      10     11  ...
------------------------------------------------------------------
  1 | 3   9    21   63    147   357   903    2499   6069   13915
  2 | 5   15   27   81    171   399   987    2709   6321   14847
  3 | 7   25   33   99    189   441   1029   2793   6325   15125
  4 | 11  35   39   117   207   483   1113   2961   6783   15141
  5 | 13  45   51   153   243   513   1197   3025   6875   15351
  6 | 17  49   55   165   261   567   1239   3087   6909   15729
  7 | 19  77   57   195   275   609   1265   3249   7011   16023
  8 | 23  91   65   231   279   621   1281   3339   7203   16611
  9 | 29  121  69   255   297   651   1375   3381   7353   16779
  10| 31  135  75   273   333   729   1407   3591   7581   17157
   ...
a(9) = 25 = T(3,3) since only 9 and 15 are smaller odd numbers whose symmetric representation of sigma consists of three parts. All 3 parts of the symmetric representation of sigma for 9 and for 25 have width 1 while the center part for that of 15 has width 2.

Cf. A065091, A237270, A237271, A237593, A239663, A247687, A320537, A341969, A341970, A341971, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax+1], i, p}, For[i=3, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax+1&&Length[list[[p]]]

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).

A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

Original entry on oeis.org

6, 12, 78, 20, 102, 1014, 24, 114, 1734, 12246, 28, 138, 2166, 12714, 171366, 40, 174, 3174, 13026, 501126, 1922622, 48, 186, 5046, 13182, 781926, 2057406, 28960854, 56, 222, 5766, 13494, 1679046, 2067546, 144825414, 300014754, 80, 246, 8214, 13962, 4243686, 2072382, 282275286, 300137214, 4174476774

Offset: 1

Hartmut F. W. Hoft, Apr 21 2024

Every number in this sequence is even since the symmetric representation of sigma for an odd number q starts 101. Each number in column m of T(n,m) has 2*m odd divisors.
Since u(m) = 2 * 3 * 13^(m-1), m>=1, has 2m odd divisors and 1 < 3 < 4 < 4*3 < 13 < 3*13 < 4*13 < 3*4*13 < 13^2 < ..., the symmetric representation of sigma for u(m) consists of m copies of unimodal pattern 121. Therefore, every column in the table T(n,m), m>=1, contains infinitely many entries. Number u(m) is the smallest entry in the m-th column when m is prime.
In general: If m>1 then T(n,m) = 2^k * q, k>=1, q odd, has at least 4 odd divisors which satisfy
d_(2i+2)  < 2^(k+1) * d_(2i+1) < 2^(k+1) * d_(2i+2) < d_(2i+3), i>=0,
with the odd divisors d_j of n in increasing order.

			a(1) = T(1,1) = 6, its symmetric representation of sigma, SRS(6), has unimodal pattern 121 and a single unit of width 2 at the diagonal.
a(3) = T(1,2) = 78, SRS(78) has unimodal pattern 1210121;
a(10) = T(1,4) = 12246, SRS(12246) has unimodal pattern 121012101210121;
both symmetric representations of sigma have width 0 at the diagonal where two parts meets.
Each number in the m-th column has 2m odd divisors. T(1,9) = 4174476774.
  -------------------------------------------------------------------------
   n\m  1    2     3     4       5         6          7          8
  -------------------------------------------------------------------------
   1|   6   78   1014  12246   171366   1922622    28960854  300014754 ...
   2|  12  102   1734  12714   501126   2057406   144825414  300137214 ...
   3|  20  114   2166  13026   781926   2067546   282275286  300235182 ...
   4|  24  138   3174  13182   1679046  2072382   888215334  300357642 ...
   5|  28  174   5046  13494   4243686  2081742  3568939926  300431118 ...
   6|  40  186   5766  13962   5541126  2091882     ...      300602562 ...
   7|  48  222   8214  14118   8487372  2097966              300651546 ...
   8|  56  246  10086  14898  11082252  2110134              300896466 ...
   9|  80  258  10092  15054  11244966  2112162              301165878 ...
  10|  88  282  11094  15366  16954566  2116218              301386306 ...
  ...

Row 1 gives A372181.
Column 1 gives A370205.
Column 2 gives A370206.
Cf. A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A341971, A342592, A342594, A342595, A342596, A367377, A370209.

divQ[k_, {d1_, d2_, d3_}] := d2<2^(k+1)d1&&2^(k+1)d2

T(n,1) = 2^k * p with odd prime p satisfying p < 2^(k+1), see A370205.

T(n,2) = 2^k * p * q, k > 0, p and q prime, 2 < p < 2^(k+1) < 2^(k+1) * p < q, see A370206.

cp's OEIS Frontend

A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

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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.

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A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

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A346969 1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 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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A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

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