A249407 -id:A249407 - cp's OEIS frontend

A249411 Oblong numbers in A249407.

2, 12, 30, 42, 72, 110, 156, 210, 272, 342, 380, 420, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832

Offset: 1

Reinhard Zumkeller, Oct 31 2014

nonn

A005369(a(n)) = 1.

			Distribution of oblong numbers on A249406 and its complement,
m*(m+1) factorizations are shown in respective columns:
.   n | A002378 | A249406 | A249407     n | A002378 | A249406 | A249407
.  ---+---------+---------+--------    ---+---------+---------+--------
.   1 |       2 |         |     1*2    13 |     182 |   13*14 |
.   2 |       6 |     2*3 |            14 |     210 |         |   14*15
.   3 |      12 |         |     3*4    15 |     240 |   15*16 |
.   4 |      20 |     4*5 |            16 |     272 |         |   16*17
.   5 |      30 |         |     5*6    17 |     306 |   17*18 |
.   6 |      42 |         |     6*7    18 |     342 |         |   18*19
.   7 |      56 |     7*8 |            19 |     380 |         |   19*20
.   8 |      72 |         |     8*9    20 |     420 |         |   20*21
.   9 |      90 |    9*10 |            21 |     462 |         |   21*22
.  10 |     110 |         |   10*11    22 |     506 |   22*23 |
.  11 |     132 |   11*12 |            23 |     552 |         |   23*24
.  12 |     156 |         |   12*13    24 |     600 |   24*25 |        .

Reinhard Zumkeller, Table of n, a(n) for n = 1..301

Cf. A249407, A005369, subsequence of A002378.

a249411 n = a249411_list !! (n-1)
a249411_list = filter ((== 1) . a005369) a249407_list

A249406 Start with a(1) = 1, and extend by the rule that the next term is the product of the two most recent non-terms of the sequence.

Original entry on oeis.org

1, 6, 20, 56, 90, 132, 182, 240, 306, 399, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8099, 8556, 8930, 9312

Offset: 1

Reinhard Zumkeller, Oct 31 2014

nonn

Compare to A075326, where not products, but sums of the two most recent non-terms are considered;
a(195) = 159200 is the smallest even term not of the form m*(m+1); see also A249408, the set of all non-oblong terms of this sequence.
a(10) = 399 is the smallest odd term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000301, A000304, A075326, A249055, A249407 (complement), subsequence of A002808.

import Data.List ((\\))
a249406 n = a249406_list !! (n-1)
a249406_list = 1 : f [2..] where
   f ws@(u:v:_) = y : f (ws \\ [u, v, y]) where y = u * v

A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Rémy Sigrist, Mar 03 2023

nonn

In other words, a(1), a(2), a(1)*a(2), a(3), a(4), a(3)*a(4), a(1)*a(2)*a(3)*a(4), a(5), a(6), a(5)*a(6), etc. are all distinct.
In particular, all terms are distinct (but not necessarily in increasing order).
We can arrange the terms of the sequence as the leaves of a perfect infinite binary tree, the products with e > 0 corresponding to parent nodes; each node will contain a different value and all values will appear in the tree (if n = 2^m+1 for some m > 0, then a(n) will equal the least value > 1 missing so far in the tree).
This sequence is a variant of A361144 where we use products instead of sums.
The data section matches that of A249407, however a(115) = 121 whereas A249407(115) = 120.

			The first terms (at the bottom of the tree) alongside the corresponding products are:
                          1067062284288000
                  ---------------------------------
               604800                        1764322560
          -----------------               -----------------
         120            5040            24024           73440
      ---------       ---------       ---------       ---------
      6      20      56      90      132     182     240     306
    -----   -----   -----   -----   -----   -----   -----   -----
    2   3   4   5   7   8   9  10  11  12  13  14  15  16  17  18

Rémy Sigrist, PARI program

Cf. A249407, A361144, A361234.

PARI
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A249411 Oblong numbers in A249407.

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A249406 Start with a(1) = 1, and extend by the rule that the next term is the product of the two most recent non-terms of the sequence.

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Haskell

A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.

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PARI

cp's OEIS Frontend

A249411 Oblong numbers in A249407.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A249406 Start with a(1) = 1, and extend by the rule that the next term is the product of the two most recent non-terms of the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A360305 Lexicographically earliest sequence of integers > 1 such that the products Product_{i = 1+k2^e..(k+1)2^e} a(i) with k, e >= 0 are all distinct.