A340581 Irregular triangle read by rows in which row n has length A014153(n-1) and every column k lists the positive integers A000027, n >= 1, k >= 1.

1, 2, 1, 1, 3, 2, 2, 1, 1, 1, 1, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

Row n lists in nonincreasing order the first A014153(n-1) terms of A176206.

In other words: row n lists in nonincreasing order the terms of the first n rows of triangle A176206.

Conjecture: all divisors of all terms in row n are also all parts of all partitions of all positive integers <= n.

The conjecture is in accordance with the conjectures in A336811 and in A176206.

A336811 contains the most elementary conjecture about the correspondence divisors/partitions.

The connection with A336811 (the main sequence) is as follows: A336811 --> A176206 --> this sequence.

Links

Table of n, a(n) for n=1..96.

Example

Triangle begins:
1;
2, 1, 1;
3, 2, 2, 1, 1, 1, 1;
4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...
For n = 4, by definition the length of row 6 is A014153(4-1) = A014153(3) = 14, so the row 4 of triangle has 14 terms. Since every column lists the positive integers A000027 so the row 4 is [4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1].
Then we have that the divisors of the numbers of the 4th row are:
.
4th row of the triangle ----------> 4  3  3  2  2  2  2  1  1  1  1  1  1  1
                                    2  1  1  1  1  1  1
                                    1
.
There are fourteen 1's, five 2's, two 3's and one 4.
In total there are 14 + 5 + 2 + 1 = 22 divisors.
On the other hand all partitions of all positive integers <= 4 are as shown below:
.
.    Partition   Partitions    Partitions     Partitions
.       of 1        of 2          of 3           of 4
.
.                                             4
.                                             2  2
.                               3             3  1
.                   2           2  1          2  1  1
.        1          1  1        1  1  1       1  1  1  1
.
In these partitions there are fourteen 1's, five 2's, two 3's and one 4.
In total there are 14 + 5 + 2 + 1 = A284870(4) = 22 parts.
Finally in accordance with the conjecture we can see that all divisors of all numbers in the 4th row of the triangle are the same positive integers as all parts of all partitions of all positive integers <= 4.

Crossrefs

Cf. A000041, A000070, A027750, A014153, A176206, A284870, A336811.

Sequence in context: A288165 A343196 A016441 * A345116 A278042 A338714

Adjacent sequences: A340578 A340579 A340580 * A340582 A340583 A340584

Keyword

nonn,tabf

Author

Omar E. Pol, Jan 14 2021

Status

approved