A299773 -id:A299773 - cp's OEIS frontend

A316886 Where records occur in A299773.

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1620, 1680, 1800, 1920, 1980

Offset: 1

Omar E. Pol, Jul 15 2018

nonn

First differs from the highly abundant numbers (A002093) at a(41) = 672, while A002093(41) = 720.

			After a(40) = 660 we have that in the sequence A299773 the terms A299773(661)..A299773(671) are less than A299773(660) = 7187172406818511650939943511021032181119077585. The next term greater than A299773(660) is A299773(672) = 7187180892191062904110726467218877665371246875, so a(41) = 672. Note that both 660 and 672 have the same number of divisors (tau(660) = tau(672) = 24) and the same sum of divisors (sigma(660) = sigma(672) = 2016).

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A000005, A000041, A000203, A002093, A299773, A316916.

More terms from Amiram Eldar, Aug 22 2019

A316916 Records in A299773.

Original entry on oeis.org

1, 2, 3, 9, 48, 119, 269, 2740, 5218, 24319, 42150, 792335, 4513067, 54782479, 101527454, 2489565187, 204017663873, 2328040254212, 26770510056270, 60003612992726, 246161422312909, 2017680047306542, 498466688538984687, 7548204089604377821, 705600340631647816172, 26237144094556522735561

Offset: 1

Omar E. Pol, Jul 16 2018

nonn

The indices of these records in A299773 first differ from the highly abundant numbers (A002093) at the 41st term, see A316886.

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A000041, A000203, A002093, A299773, A316886.

A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 5, 1, 7, 1, 8, 10, 11, 1, 15, 1, 16, 21, 22, 1, 27, 30, 1, 31, 41, 42, 1, 56, 1, 57, 69, 73, 76, 77, 1, 101, 1, 102, 134, 135, 1, 160, 172, 176, 1, 177, 221, 230, 231, 1, 297, 1, 298, 353, 380, 384, 385, 1, 490, 1, 491, 604, 615, 626, 627, 1

Offset: 1

Omar E. Pol, Mar 29 2018

Note that n is one of the partitions of n into equal parts.
If n is even then row n ending in [p(n) - 1, p(n)], where p(n) = A000041(n).
T(n,k) > p(n - 1), if 1 < k <= A000005(n).
Removing the 1's then all terms of the sequence are in increasing order.
If n is even then row n starts with [1, p(n - 1) + 1]. - David A. Corneth and Omar E. Pol, Aug 26 2018

			Triangle begins:
  1;
  1,   2;
  1,   3;
  1,   4,   5;
  1,   7;
  1,   8,  10,  11;
  1,  15;
  1,  16,  21,  22;
  1,  27,  30;
  1,  31,  41,  42;
  1,  56;
  1,  57,  69,  73,  76,  77;
  1, 101;
  1, 102, 134, 135;
  1, 160, 172, 176;
  ...
For n = 6 the partitions of 6 into equal parts are [1, 1, 1, 1, 1, 1], [2, 2, 2], [3, 3] and [6]. Then we have that in the list of colexicographically ordered partitions of 6 these partitions are in the rows 1, 8, 10 and 11 respectively as shown below, so the 6th row of the triangle is [1, 8, 10, 11].
-------------------------------------------------------------
   p      Diagram        Partitions of 6
-------------------------------------------------------------
        _ _ _ _ _ _
   1   |_| | | | | |    [1, 1, 1, 1, 1, 1]  <--- equal parts
   2   |_ _| | | | |    [2, 1, 1, 1, 1]
   3   |_ _ _| | | |    [3, 1, 1, 1]
   4   |_ _|   | | |    [2, 2, 1, 1]
   5   |_ _ _ _| | |    [4, 1, 1]
   6   |_ _ _|   | |    [3, 2, 1]
   7   |_ _ _ _ _| |    [5, 1]
   8   |_ _|   |   |    [2, 2, 2]  <--- equal parts
   9   |_ _ _ _|   |    [4, 2]
  10   |_ _ _|     |    [3, 3]  <--- equal parts
  11   |_ _ _ _ _ _|    [6]  <--- equal parts
.

Row n has length A000005(n).
Right border gives A000041, n >= 1.
Column 1 gives A000012.
Records give A317296.
Cf. A211992 (partitions in colexicographic order).
Cf. A027750, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299775.

row(n) = {if(n == 1, return([1])); my(nd = numdiv(n), res = vector(nd)); res[1] = 1; res[nd] = numbpart(n); if(nd > 2, t = nd - 1; p = vecsort(partitions(n)); forstep(i = #p - 1, 2, -1, if(p[i][1] == p[i][#p[i]], res[t] = i; t--; if(t==1, return(res)))), return(res))} \\ David A. Corneth, Aug 17 2018

Terms a(46) and beyond from David A. Corneth, Aug 16 2018

A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 7, 6, 11, 14, 15, 22, 25, 29, 30, 25, 42, 55, 56

Offset: 1

Omar E. Pol, Mar 29 2018

If n > 1 and n is odd then row n ending in [p(n) - 1, p(n)], where p(n) is A000041(n).

			Triangle begins:
   1;
   2;
   2,  3;
   5;
   6,  7;
   6, 11;
  14, 15;
  22;
  25, 29, 30;
  25, 42;
  55, 56;
...
For n = 9 the partitions of 9 into consecutive parts are [4, 3, 2], [5, 4] and [9]. Then we have that in the list of colexicographically ordered partitions of 9 these partitions are in the rows 25, 29 and 30 respectively as shown below, so the 9th row of the triangle is [25, 29, 30].
--------------------------------------------------------
   p         Diagram          Partitions of 9
--------------------------------------------------------
        1 2 3 4 5 6 7 8 9
        _ _ _ _ _ _ _ _ _
   1   |_| | | | | | | | |   [1, 1, 1, 1, 1, 1, 1, 1, 1]
   2   |_ _| | | | | | | |   [2, 1, 1, 1, 1, 1, 1, 1]
   3   |_ _ _| | | | | | |   [3, 1, 1, 1, 1, 1, 1]
   4   |_ _|   | | | | | |   [2, 2, 1, 1, 1, 1, 1]
   5   |_ _ _ _| | | | | |   [4, 1, 1, 1, 1, 1]
   6   |_ _ _|   | | | | |   [3, 2, 1, 1, 1, 1]
   7   |_ _ _ _ _| | | | |   [5, 1, 1, 1, 1]
   8   |_ _|   |   | | | |   [2, 2, 2, 1, 1, 1]
   9   |_ _ _ _|   | | | |   [4, 2, 1, 1, 1]
  10   |_ _ _|     | | | |   [3, 3, 1, 1, 1]
  11   |_ _ _ _ _ _| | | |   [6, 1, 1, 1]
  12   |_ _ _|   |   | | |   [3, 2, 2, 1, 1]
  13   |_ _ _ _ _|   | | |   [5, 2, 1, 1]
  14   |_ _ _ _|     | | |   [4, 3, 1, 1]
  15   |_ _ _ _ _ _ _| | |   [7, 1, 1]
  16   |_ _|   |   |   | |   [2, 2, 2, 2, 1]
  17   |_ _ _ _|   |   | |   [4, 2, 2, 1]
  18   |_ _ _|     |   | |   [3, 3, 2, 1]
  19   |_ _ _ _ _ _|   | |   [6, 2, 1]
  20   |_ _ _ _ _|     | |   [5, 3, 1]
  21   |_ _ _ _|       | |   [4, 4, 1]
  22   |_ _ _ _ _ _ _ _| |   [8, 1]
  23   |_ _ _|   |   |   |   [3, 2, 2, 2]
  24   |_ _ _ _ _|   |   |   [5, 2, 2]
  25   |_ _ _ _|     |   |   [4, 3, 2]   <--- Consecutive parts
  26   |_ _ _ _ _ _ _|   |   [7, 2]
  27   |_ _ _|     |     |   [3, 3, 3]
  28   |_ _ _ _ _ _|     |   [6, 3]
  29   |_ _ _ _ _|       |   [5, 4]   <--- Consecutive parts
  30   |_ _ _ _ _ _ _ _ _|   [9]   <--- Consecutive parts
.

Row n has length A001227(n).
Right border gives A000041, n >= 1.
Cf. A211992 (partitions in colexicographic order).
Cf. A299765 (partitions into consecutive parts).
For tables of partitions into consecutive parts see also A286000 and A286001.
Cf. A000041, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299774.

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A316886 Where records occur in A299773.

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A316916 Records in A299773.

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A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.

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A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

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