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.

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

Author

Omar E. Pol, Sep 20 2015

Keywords

• nonn
• tabf

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