cp's OEIS Frontend

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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A261699 Triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists positive terms interleaved with k-1 zeros, starting in row k(k+1)/2. If k is odd the positive terms of column k are k's, otherwise if k is even the positive terms of column k are the odd numbers greater than k in increasing order.

Original entry on oeis.org

1, 1, 1, 3, 1, 0, 1, 5, 1, 0, 3, 1, 7, 0, 1, 0, 0, 1, 9, 3, 1, 0, 0, 5, 1, 11, 0, 0, 1, 0, 3, 0, 1, 13, 0, 0, 1, 0, 0, 7, 1, 15, 3, 0, 5, 1, 0, 0, 0, 0, 1, 17, 0, 0, 0, 1, 0, 3, 9, 0, 1, 19, 0, 0, 0, 1, 0, 0, 0, 5, 1, 21, 3, 0, 0, 7, 1, 0, 0, 11, 0, 0, 1, 23, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 25, 0, 0, 5, 0, 1, 0, 0, 13, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Sep 20 2015

Keywords

Comments

Conjecture: the positive terms in row n are the odd divisors of n.
Note that the elements appear with an unusual ordering, for example; row 45 is 1, 45, 3, 0, 5, 15, 0, 0, 9.
The positive terms give A261697.
Row n has length A003056(n) hence column k starts in row A000217(k).
The number of positive terms in row n is A001227(n).
The sum of row n is A000593(n).
The connection with the symmetric representation of sigma is as follows: A237048 --> A235791 --> A237591 --> A237593.
Proof of the conjecture: let n = 2^m*s*t with s and t odd. The property stated in A237048 verifies the conjecture with odd divisor k <= A003056(n) of n in position k and odd divisor t > A003056(n) in position 2^(m+1)*s. Therefore reading in row n the nonzero odd positions from left to right and then the nonzero even positions from right to left gives all odd divisors of n in increasing order. - Hartmut F. W. Hoft, Oct 25 2015
A237048 gives the signum function (A057427) of this sequence. - Omar E. Pol, Nov 14 2016
From Peter Munn, Jul 30 2017: (Start)
Each odd divisor d of n corresponds to n written as a sum of consecutive integers (n/d - (d-1)/2) .. (n/d + (d-1)/2). After canceling any corresponding negative and positive terms and deleting any zero term, the lower bound becomes abs(n/d - d/2) + 1/2, leaving k terms where k = n/d + d/2 - abs(n/d - d/2). It can be shown T(n,k) = d.
This sequence thereby defines a one to one relationship between odd divisors of n and partitions of n into k consecutive parts.
The relationship is expressed below using 4 sequences (with matching row lengths), starting with this one:
A261699(n,k) = d, the odd divisor.
A211343(n,k) = abs(n/d - d/2) + 1/2, smallest part.
A285914(n,k) = k, number of parts.
A286013(n,k) = n/d + (d-1)/2, largest part.
If no partition of n into k consecutive parts exists, the corresponding sequence terms are 0.
(End)

Examples

			Triangle begins:
1;
1;
1,  3;
1,  0;
1,  5;
1,  0,  3;
1,  7,  0;
1,  0,  0;
1,  9,  3;
1,  0,  0,  5;
1, 11,  0,  0;
1,  0,  3,  0;
1, 13,  0,  0;
1,  0,  0,  7;
1, 15,  3,  0,  5;
1,  0,  0,  0,  0;
1, 17,  0,  0,  0;
1,  0,  3,  9,  0;
1, 19,  0,  0,  0;
1,  0,  0,  0,  5;
1, 21,  3,  0,  0,  7;
1,  0,  0, 11,  0,  0;
1, 23,  0,  0,  0,  0;
1,  0,  3,  0,  0,  0;
1, 25,  0,  0,  5,  0;
1,  0,  0, 13,  0,  0;
1, 27,  3,  0,  0,  9;
1,  0,  0,  0,  0,  0,  7;
...
From _Omar E. Pol_, Dec 19 2016: (Start)
Illustration of initial terms in a right triangle whose structure is the same as the structure of A237591:
Row                                                         _
1                                                         _|1|
2                                                       _|1 _|
3                                                     _|1  |3|
4                                                   _|1   _|0|
5                                                 _|1    |5 _|
6                                               _|1     _|0|3|
7                                             _|1      |7  |0|
8                                           _|1       _|0 _|0|
9                                         _|1        |9  |3 _|
10                                      _|1         _|0  |0|5|
11                                    _|1          |11  _|0|0|
12                                  _|1           _|0  |3  |0|
13                                _|1            |13   |0 _|0|
14                              _|1             _|0   _|0|7 _|
15                            _|1              |15   |3  |0|5|
16                          _|1               _|0    |0  |0|0|
17                        _|1                |17    _|0 _|0|0|
18                      _|1                 _|0    |3  |9  |0|
19                    _|1                  |19     |0  |0 _|0|
20                  _|1                   _|0     _|0  |0|5 _|
21                _|1                    |21     |3   _|0|0|7|
22              _|1                     _|0      |0  |11 |0|0|
23            _|1                      |23      _|0  |0  |0|0|
24          _|1                       _|0      |3    |0 _|0|0|
25        _|1                        |25       |0   _|0|5  |0|
26      _|1                         _|0       _|0  |13 |0 _|0|
27    _|1                          |27       |3    |0  |0|9 _|
28   |1                            |0        |0    |0  |0|0|7|
... (End)

Crossrefs

Cf. A000217, A000593, A001227, A003056, A005408, A027750, A057427, A182469, A196020, A211343, A236104, A235791, A236112, A237048, A237591, A237593, A261350, A261697, A261698, A285914, A286013.

Programs

  • Mathematica
    T[n_, k_?OddQ] /; n == k (k + 1)/2 := k; T[n_, k_?OddQ] /; Mod[n - k (k + 1)/2, k] == 0 := k; T[n_, k_?EvenQ] /; n == k (k + 1)/2 := k + 1; T[n_, k_?EvenQ] /; Mod[n - k (k + 1)/2, k] == 0 := T[n - k, k] + 2; T[, ] = 0; Table[T[n, k], {n, 1, 26}, {k, 1, Floor[(Sqrt[1 + 8 n] - 1)/2]}] // Flatten (* Jean-François Alcover, Sep 21 2015 *)
(* alternate definition using function a237048 *)
T[n_, k_] := If[a237048[n, k] == 1, If[OddQ[k], k, 2n/k], 0] (* Hartmut F. W. Hoft, Oct 25 2015 *)

Formula

From Hartmut F. W. Hoft, Oct 25 2015: (Start)
T(n, k) = 2n/k, if A237048(n, k) = 1 and k even,
and in accordance with the definition:
T(n, k) = k, if A237048(n, k) = 1 and k odd,
T(n, k) = 0 otherwise; for k <= A003056(n).
(End)
For m >= 1, d >= 1 and odd, T(m*d, m + d/2 - abs(m - d/2)) = d. - Peter Munn, Jul 24 2017

A266531 Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 6, 18, 15, 16, 7, 25, 21, 81, 32, 10, 36, 27, 162, 45, 64, 11, 49, 30, 324, 63, 729, 128, 12, 50, 33, 625, 75, 1458, 105, 256, 13, 72, 35, 648, 90, 2916, 135, 225, 512, 14, 98, 39, 1250, 99, 5832, 165, 441, 405, 1024, 17, 100, 42, 1296, 117, 11664, 189, 450, 567, 59049, 2048, 19, 121, 51, 2401, 126, 15625
Offset: 1

Views

Author

Omar E. Pol, Apr 02 2016

Keywords

Comments

T(n,k) is the n-th positive integer with exactly k odd divisors.
This is a permutation of the natural numbers.
T(n,k) is also the n-th number j with the property that the symmetric representation of sigma(j) has k subparts (cf. A279387). - Omar E. Pol, Dec 27 2016
T(n,k) is also the n-th positive integer with exactly k partitions into consecutive parts. - Omar E. Pol, Aug 16 2018

Examples

			The corner of the square array begins:
    1,  3,  9, 15,   81,  45,   729, 105,  225,  405, ...
    2,  5, 18, 21,  162,  63,  1458, 135,  441,  567, ...
    4,  6, 25, 27,  324,  75,  2916, 165,  450,  810, ...
    8,  7, 36, 30,  625,  90,  5832, 189,  882,  891, ...
   16, 10, 49, 33,  648,  99, 11664, 195,  900, 1053, ...
   32, 11, 50, 35, 1250, 117, 15625, 210, 1089, 1134, ...
   64, 12, 72, 39, 1296, 126, 23328, 231, 1225, 1377, ...
  128, 13, 98, 42, 2401, 147, 31250, 255, 1521, 1539, ...
  ...

Links

Crossrefs

Row 1 is A038547.
Column 1-10: A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, A267892, A267893.
Cf. A001227, A182469, A236104, A237591, A237593, A240062, A261697, A261698, A261699, A279387, A286000, A286001, A296508.

A261697 Irregular triangle read by rows in which row n lists the odd divisors of n in the ordering given by A261699.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 9, 3, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 15, 3, 5, 1, 1, 17, 1, 3, 9, 1, 19, 1, 5, 1, 21, 3, 7, 1, 11, 1, 23, 1, 3, 1, 25, 5, 1, 13, 1, 27, 3, 9, 1, 7, 1, 29, 1, 3, 15, 5, 1, 31, 1, 1, 33, 3, 11, 1, 17, 1, 35, 5, 7, 1, 3, 9, 1, 37, 1, 19, 1, 39, 3, 13, 1, 5, 1, 41, 1, 3, 21, 7, 1, 43, 1, 11, 1, 45, 3, 5, 15, 9
Offset: 1

Views

Author

Omar E. Pol, Sep 21 2015

Keywords

Comments

Positive terms of A261699.
Note that this ordering is unusual.
Row lengths give A001227.
Row sums give A000593.
Another version of A182469 from which differs at a(14), or T(9,2).
For a similar version see A261698 from which differs at a(34), or T(18,2).

Examples

			List of divisors of 45 from distinct sequences:
45th row of triangle A182469: 1, 3, 5, 9, 15, 45.
45th row of triangle A261698: 1, 45, 3, 15, 5, 9.
45th row of this triangle...: 1, 45, 3, 5, 15, 9.
Triangle begins:
  1;
  1;
  1,  3;
  1;
  1,  5;
  1,  3;
  1,  7;
  1;
  1,  9,  3;
  1,  5;
  1, 11;
  1,  3;
  1, 13;
  1,  7;
  1, 15,  3,  5;
  1;
  1, 17;
  1,  3,  9;
  1, 19;
  1,  5;
  1, 21,  3,  7;
  ...

Crossrefs

Cf. A000593, A001227, A027750, A182469, A261698, A261699.

A379634 Irregular triangle read by rows in which row n lists the odd divisors of n ordered as the mirror of A261697.

Original entry on oeis.org

1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 5, 3, 15, 1, 1, 17, 1, 9, 3, 1, 19, 1, 5, 1, 7, 3, 21, 1, 11, 1, 23, 1, 3, 1, 5, 25, 1, 13, 1, 9, 3, 27, 1, 7, 1, 29, 1, 5, 15, 3, 1, 31, 1, 1, 11, 3, 33, 1, 17, 1, 7, 5, 35, 1, 9, 3, 1, 37, 1, 19, 1, 13, 3, 39, 1, 5, 1, 41, 1, 7, 21, 3, 1, 43, 1, 11, 1
Offset: 1

Views

Author

Omar E. Pol, Dec 31 2024

Keywords

Comments

Row n gives the last A001227(n) terms of the n-th row of A379630 and of A379631.
For a correspondence between the row n and the partitions of n into consecutive parts see A379630.

Examples

			Triangle begins:
   1;
   1;
   3,  1;
   1;
   5,  1;
   3,  1;
   7,  1;
   1;
   3,  9,  1;
   5,  1;
  11,  1;
   3,  1;
  13,  1;
   7,  1;
   5,  3, 15,  1;
   1;
  17,  1;
   9,  3,  1;
  19,  1;
   5,  1;
   7,  3, 21,  1;
  11,  1;
  23,  1;
   3,  1;
   5, 25,  1;
  13,  1;
   9,  3, 27,  1;
   7,  1;
  ...
Illustration of initial terms:
   Row    _
   1     |1|_
   2     |_ 1|_
   3     |3|  1|_
   4     | |_   1|_
   5     |_ 5|    1|_
   6     |3| |_     1|_
   7     | |  7|      1|_
   8     | |_  |_       1|_
   9     |_ 3|  9|        1|_
  10     |5| |   |_         1|_
  11     | | |_  11|          1|_
  12     | |  3|   |_           1|_
  13     | |_  |   13|            1|_
  14     |_ 7| |_    |_             1|_
  15     |5| |  3|   15|              1|_
  16     | | |   |     |_               1|_
  17     | | |_  |_    17|                1|_
  18     | |  9|  3|     |_                 1|_
  19     | |_  |   |     19|                  1|_
  20     |_ 5| |   |_      |_                   1|_
  21     |7| | |_   3|     21|                    1|_
  22     | | | 11|   |       |_                     1|_
  23     | | |   |   |_      23|                      1|_
  24     | | |_  |    3|       |_                       1|_
  25     | |  5| |_    |       25|                        1|_
  26     | |_  | 13|   |_        |_                          1|_
  27     |_ 9| |   |    3|       27|                           1|_
  28     |7| | |   |     |         |                             1|
  ...
The diagram is also the right part of the diagram of A379630 and of A379631.
The geometrical structure is the same as the diagram of A261350 which is the mirror of A237591.

Crossrefs

Mirror of A261697.
Right border gives A000012.
Row lengths give A001227.
Row sums give A000593.
Other versions are A182469, A261697, A261698.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A261350, A261699, A379630, A379631, A379632, A379633.
Showing 1-4 of 4 results.

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