A341309
Sum of odd divisors of n that are <= A003056(n).
Original entry on oeis.org
1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 8, 1, 9, 1, 1, 4, 1, 13, 4, 1, 1, 4, 6, 1, 11, 1, 1, 18, 1, 1, 4, 8, 6, 4, 1, 1, 13, 6, 8, 4, 1, 1, 9, 1, 1, 20, 1, 6, 15, 1, 1, 4, 13, 1, 13, 1, 1, 9, 1, 19, 4, 1, 6, 13, 1, 1, 11, 6, 1, 4, 12, 1, 18, 21
Offset: 1
-
A341309[n_]:=With[{t=Floor[(Sqrt[8n+1]-1)/2]},DivisorSum[n,#&,OddQ[#]&&#<=t&]];
Array[A341309,100] (* Paolo Xausa, Mar 25 2023 *)
-
a(n) = my(m=n>>valuation(n, 2), s=(sqrtint(8*n+1)-1)\2); sumdiv(m, d, if (d <= s, d)); \\ Michel Marcus, Mar 25 2023
A379630
Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines.
Original entry on oeis.org
1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 5, 2, 5, 1, 6, 1, 3, 1, 7, 3, 7, 1, 8, 1, 9, 4, 2, 3, 9, 1, 10, 1, 5, 1, 11, 5, 11, 1, 12, 3, 3, 1, 13, 6, 13, 1, 14, 2, 7, 1, 15, 7, 4, 1, 5, 3, 15, 1, 16, 1, 17, 8, 17, 1, 18, 5, 3, 9, 3, 1, 19, 9, 19, 1, 20, 2, 5, 1, 21, 10, 6, 1, 7, 3, 21, 1, 22, 4, 11, 1, 23, 11, 23, 1, 24, 7, 3, 1
Offset: 1
Triangle begins:
1, 1;
2, 1;
3, 1, 3, 1;
4, 1;
5, 2, 5, 1;
6, 1, 3, 1;
7, 3, 7, 1;
8, 1;
9, 4, 2, 3, 9, 1;
10, 1, 5, 1;
11, 5, 11, 1;
12, 3, 3, 1;
13, 6, 13, 1;
14, 2, 7, 1;
15, 7, 4, 1, 5, 3, 15, 1;
16, 1;
17, 8, 17, 1;
18, 5, 3, 9, 3, 1;
19, 9, 19, 1;
20, 2, 5, 1;
21, 10, 6, 1, 7, 3, 21, 1;
...
For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1].
On the other hand the odd divisors of 21 are [1, 3, 7, 21].
To determine how these partitions are related to the odd divisors we follow the two rules of the theorem as shown below:
The first partition is [21] and the number of parts is 1 and 1 is odd so the corresponding odd divisor of 21 is 1.
The second partition is [11, 10] and the number of parts is 2 and 2 even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21.
The third partition is [8, 7, 6] and the number of parts is 3 and 3 is odd so the corresponding odd divisor of 21 is 3.
The fourth partition is [6, 5, 4, 3, 2, 1] and the number of parts is 6 and 6 is even so the corresponding odd divisor of 21 is equal to 6 + 1 = 5 + 2 = 4 + 3 = 7.
Summarizing in a table:
--------------------------------------
Correspondence
--------------------------------------
Partitions of 21 Odd
into consecutive divisors
parts of 21
------------------- ----------
[21] .................... 1
[11, 10] ................ 21
[8, 7, 6] ................ 3
[6, 5, 4, 3, 2, 1] ....... 7
.
Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the smallest parts at the top as shown below:
------------------------------------------
Conjugate correspondence
------------------------------------------
Partitions of 21 Odd
into consecutive divisors
parts as columns of 21
------------------- ------------------
21 10 6 1 7 3 21 1
| 11 7 2 | | | |
| | 8 3 | | | |
| | | 4 | | | |
| | | 5 | | | |
| | | 6 | | | |
| | | |_______| | | |
| | |_________________| | |
| |___________________________| |
|_____________________________________|
.
Then removing all rows except the first row we have a table of conjugate correspondence for smallest parts and odd divisors as shown below:
------------------- ------------------
Smallest parts Odd divisors
------------------- ------------------
21 10 6 1 7 3 21 1
| | | |_______| | | |
| | |_________________| | |
| |___________________________| |
|_____________________________________|
.
So the 21st row of the triangle is [21, 10, 6, 1, 7, 3, 21, 1].
.
Illustration of initial terms in an isosceles triangle demonstrating the theorem:
. _ _
_|1|1|_
_|2 _|_ 1|_
_|3 |1|3| 1|_
_|4 _| | |_ 1|_
_|5 |2 _|_ 5| 1|_
_|6 _| |1|3| |_ 1|_
_|7 |3 | | | 7| 1|_
_|8 _| _| | |_ |_ 1|_
_|9 |4 |2 _|_ 3| 9| 1|_
_|10 _| | |1|5| | |_ 1|_
_|11 |5 _| | | | |_ 11| 1|_
_|12 _| |3 | | | 3| |_ 1|_
_|13 |6 | _| | |_ | 13| 1|_
_|14 _| _| |2 _|_ 7| |_ |_ 1|_
_|15 |7 |4 | |1|5| | 3| 15| 1|_
_|16 _| | | | | | | | |_ 1|_
_|17 |8 _| _| | | | |_ |_ 17| 1|_
_|18 _| |5 |3 | | | 9| 3| |_ 1|_
_|19 |9 | | _| | |_ | | 19| 1|_
_|20 _| _| | |2 _|_ 5| | |_ |_ 1|_
|21 |10 |6 | | |1|7| | | 3| 21| 1|
.
The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangle A286001.
Note that the diagram also can be interpreted as a template which after folding gives a 90 degree pop-up card which has essentially the same structure as the stepped pyramid described in A245092.
.
Cf.
A196020,
A204217,
A235791,
A236104,
A237048,
A237270,
A237271,
A237591,
A237593,
A245092,
A262626,
A341971.
A352425
Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 3, 2, 1, 7, 8, 9, 4, 3, 2, 10, 11, 12, 5, 4, 3, 13, 14, 15, 6, 5, 4, 5, 4, 3, 2, 1, 16, 17, 18, 7, 6, 5, 19, 20, 6, 5, 4, 3, 2, 21, 8, 7, 6, 22, 23, 24, 9, 8, 7, 25, 7, 6, 5, 4, 3, 26, 27, 10, 9, 8, 28, 7, 6, 5, 4, 3, 2, 1, 29, 30, 11, 10, 9, 8, 7, 6, 5, 4
Offset: 1
Triangle begins:
[1];
[2];
[3],
[4];
[5];
[6], [3, 2, 1];
[7];
[8];
[9], [4, 3, 2];
[10];
[11];
[12], [5, 4, 3];
[13];
[14];
[15], [6, 5, 4], [5, 4, 3, 2, 1];
[16];
[17];
[18], [7, 6, 5];
[19];
[20], [6, 5, 4, 3, 2];
[21], [8, 7, 6];
[22];
[23];
[24], [9, 8, 7];
[25], [7, 6, 5, 4, 3];
[26];
[27], [10, 9, 8];
[28], [7, 6, 5, 4, 3, 2, 1];
...
In the diagram below the m-th staircase walk starts at row A000384(m).
The number of horizontal line segments in the n-th row equals A082647(n), the number of partitions of n into an odd number of consecutive parts, so we can find such partitions as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [6, 5, 4]. [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
_
_|1|
_|2 |
_|3 |
_|4 |
_|5 _|
_|6 |3|
_|7 |2|
_|8 _|1|
_|9 |4 |
_|10 |3 |
_|11 _|2 |
_|12 |5 |
_|13 |4 |
_|14 _|3 _|
_|15 |6 |5|
_|16 |5 |4|
_|17 _|4 |3|
_|18 |7 |2|
_|19 |6 _|1|
_|20 _|5 |6 |
_|21 |8 |5 |
_|22 |7 |4 |
_|23 _|6 |3 |
_|24 |9 _|2 |
_|25 |8 |7 |
_|26 _|7 |6 |
_|27 |10 |5 _|
|28 |9 |4 |7|
...
The diagram is infinite.
For more information about the diagram see A286000.
Row n contains
A082647(n) of the mentioned partitions.
Cf.
A000384,
A003056,
A067742,
A204217,
A237048,
A237591,
A237593,
A240542,
A245092,
A285574,
A285901,
A286000,
A286001,
A320051,
A320137,
A320142,
A341309,
A351824.
A379631
Irregular triangle read by rows: T(n,m), n >= 1, m >= 1, in which row n lists the largest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem described in A379630.
Original entry on oeis.org
1, 1, 2, 1, 3, 2, 3, 1, 4, 1, 5, 3, 5, 1, 6, 3, 3, 1, 7, 4, 7, 1, 8, 1, 9, 5, 4, 3, 9, 1, 10, 4, 5, 1, 11, 6, 11, 1, 12, 5, 3, 1, 13, 7, 13, 1, 14, 5, 7, 1, 15, 8, 6, 5, 5, 3, 15, 1, 16, 1, 17, 9, 17, 1, 18, 7, 6, 9, 3, 1, 19, 10, 19, 1, 20, 6, 5, 1, 21, 11, 8, 6, 7, 3, 21, 1, 22, 7, 11, 1, 23, 12, 23, 1, 24, 9, 3, 1
Offset: 1
Triangle begins:
1, 1;
2, 1;
3, 2, 3, 1;
4, 1;
5, 3, 5, 1;
6, 3, 3, 1;
7, 4, 7, 1;
8, 1;
9, 5, 4, 3, 9, 1,
10, 4, 5, 1;
11, 6, 11, 1;
12, 5, 3, 1;
13, 7, 13, 1;
14, 5, 7, 1;
15, 8, 6, 5, 5, 3, 15, 1;
16, 1;
17, 9, 17, 1;
18, 7, 6, 9, 3, 1;
19, 10, 19, 1;
20, 6, 5, 1;
21, 11, 8, 6, 7, 3, 21, 1;
...
For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1].
On the other hand the odd divisors of 21 are [1, 3, 7, 21].
To determine how these partitions are related to the odd divisors we follow the two rules of the theorem described in A379630 as shown below:
The first partition is [21] and the number of parts is 1 and 1 is odd so the corresponding odd divisor of 21 is 1.
The second partition is [11, 10] and the number of parts is 2 and 2 is even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21.
The third partition is [8, 7, 6] and the number of parts is 3 and 3 is odd so the corresponding odd divisor of 21 is 3.
The fourth partition is [6, 5, 4, 3, 2, 1] and the number of parts is 6 and 6 is even so the corresponding odd divisor of 21 is equal to 6 + 1 = 5 + 2 = 4 + 3 = 7.
Summarizing in a table:
--------------------------------------
Correspondence
--------------------------------------
Partitions of 21 Odd
into consecutive divisors
parts of 21
------------------- ----------
[21] .................... 1
[11, 10] ................ 21
[8, 7, 6] ................ 3
[6, 5, 4, 3, 2, 1] ....... 7
.
Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the largest parts at the top as shown below:
------------------------------------------
Conjugate correspondence
------------------------------------------
Partitions of 21 Odd
into consecutive divisors
parts as columns of 21
------------------- ------------------
21 11 8 6 7 3 21 1
| 10 7 5 | | | |
| | 6 4 | | | |
| | | 3 | | | |
| | | 2 | | | |
| | | 1 | | | |
| | | |_______| | | |
| | |_________________| | |
| |___________________________| |
|_____________________________________|
.
Then removing all rows except the first row we have a table of conjugate correspondence for largest parts and odd divisors as shown below:
------------------- ------------------
Largest parts Odd divisors
------------------- ------------------
21 11 8 6 7 3 21 1
| | | |_______| | | |
| | |_________________| | |
| |___________________________| |
|_____________________________________|
.
So the 21st row of the triangle is [21, 11, 8, 6, 7, 3, 21, 1].
.
Illustration of initial terms in an isosceles triangle demonstrating the theorem described in A379630:
. _ _
_|1|1|_
_|2 _|_ 1|_
_|3 |2|3| 1|_
_|4 _| | |_ 1|_
_|5 |3 _|_ 5| 1|_
_|6 _| |3|3| |_ 1|_
_|7 |4 | | | 7| 1|_
_|8 _| _| | |_ |_ 1|_
_|9 |5 |4 _|_ 3| 9| 1|_
_|10 _| | |4|5| | |_ 1|_
_|11 |6 _| | | | |_ 11| 1|_
_|12 _| |5 | | | 3| |_ 1|_
_|13 |7 | _| | |_ | 13| 1|_
_|14 _| _| |5 _|_ 7| |_ |_ 1|_
_|15 |8 |6 | |5|5| | 3| 15| 1|_
_|16 _| | | | | | | | |_ 1|_
_|17 |9 _| _| | | | |_ |_ 17| 1|_
_|18 _| |7 |6 | | | 9| 3| |_ 1|_
_|19 |10 | | _| | |_ | | 19| 1|_
_|20 _| _| | |6 _|_ 5| | |_ |_ 1|_
|21 |11 |8 | | |6|7| | | 3| 21| 1|
.
The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangles A286000 and A379633.
Note that the diagram also can be interpreted as a template which after folding gives a 90 degree pop-up card which has essentially the same structure as the stepped pyramid described in A245092.
.
For another version with smallest parts see
A379630.
The partitions of n into consecutive parts are in the n-th row of
A299765. See also
A286000.
Cf.
A196020,
A204217,
A235791,
A236104,
A237048,
A237270,
A237271,
A237591,
A237593,
A245092,
A262626,
A379632,
A379633,
A379634.
A328368
Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.
Original entry on oeis.org
1, 2, 3, 2, 4, 2, 5, 4, 6, 4, 3, 7, 6, 3, 8, 6, 3, 9, 8, 6, 10, 8, 6, 4, 11, 10, 6, 4, 12, 10, 9, 4, 13, 12, 9, 4, 14, 12, 9, 8, 15, 14, 12, 8, 5, 16, 14, 12, 8, 5, 17, 16, 12, 8, 5, 18, 16, 15, 12, 5, 19, 18, 15, 12, 5, 20, 18, 15, 12, 10, 21, 20, 18, 12, 10, 6, 22, 20, 18, 16, 10, 6, 23, 22, 18, 16, 10, 6
Offset: 1
Triangle begins:
1;
2;
3, 2;
4, 2;
5, 4;
6, 4, 3;
7, 6, 3;
8, 6, 3;
9, 8, 6;
10, 8, 6, 4;
11, 10, 6, 4;
12, 10, 9, 4;
13, 12, 9, 4;
14, 12, 9, 8;
15, 14, 12, 8, 5;
16, 14, 12, 8, 5;
17, 16, 12, 8, 5;
18, 16, 15, 12, 5;
19, 18, 15, 12, 5;
20, 18, 15, 12, 10;
21, 20, 18, 12, 10, 6;
22, 20, 18, 16, 10, 6;
23, 22, 18, 16, 10, 6;
24, 22, 21, 16, 10, 6;
25, 24, 21, 16, 15, 6;
26, 24, 21, 20, 15, 6;
27, 26, 24, 20, 15, 12;
28, 26, 24, 20, 15, 12, 7;
...
Column k starts with k in the row
A000217(k).
Cf.
A052928,
A196020,
A204217,
A211343,
A235791,
A236104,
A235791,
A237048,
A237591,
A237593,
A245579,
A262612,
A285900,
A285914,
A285891,
A286000,
A286001,
A286013,
A299765,
A328361,
A328365,
A328371.
-
tt(n, k) = k*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019
A329321
a(n) is the total number of odd parts in all partitions of n into consecutive parts.
Original entry on oeis.org
1, 0, 2, 0, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 6, 0, 2, 4, 2, 2, 6, 2, 2, 2, 5, 2, 6, 4, 2, 6, 2, 0, 6, 2, 8, 6, 2, 2, 6, 2, 2, 8, 2, 4, 14, 2, 2, 2, 5, 4, 6, 4, 2, 8, 10, 4, 6, 2, 2, 8, 2, 2, 14, 0, 10, 10, 2, 4, 6, 8, 2, 6, 2, 2, 14, 4, 10, 10, 2, 2, 11, 2, 2, 10, 10, 2, 6, 6, 2, 16
Offset: 1
For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are six odd parts, they are [15, 7, 5, 5, 3, 1], so a(15) = 6.
Cf.
A204217 (total number of parts).
Cf.
A000079,
A001227,
A002131,
A237048,
A245579,
A285898,
A285914,
A286000,
A286001,
A299765,
A328361,
A328365,
A329322.
A329322
a(n) is the total number of even parts in all partitions of n into consecutive parts.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 3, 5, 1, 1, 4, 1, 4, 6, 3, 1, 2, 3, 3, 6, 4, 1, 7, 1, 1, 6, 3, 7, 6, 1, 3, 6, 4, 1, 7, 1, 5, 12, 3, 1, 2, 5, 6, 6, 5, 1, 9, 8, 4, 6, 3, 1, 9, 1, 3, 14, 1, 8, 9, 1, 5, 6, 9, 1, 7, 1, 3, 13, 5, 11, 10, 1, 4, 10, 3, 1, 9, 8, 3, 6, 6, 1, 18
Offset: 1
For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are five even parts, they are [8, 6, 4, 4, 2], so a(15) = 5.
Cf.
A204217 (total number of parts).
Cf.
A001227,
A237048,
A245579,
A285898,
A285914,
A286000,
A286001,
A299765,
A328361,
A328365,
A329321.
A357618
a(n) = sum of lengths of partitions of more than one consecutive positive integer adding up to n.
Original entry on oeis.org
0, 0, 0, 2, 0, 2, 3, 2, 0, 5, 4, 2, 3, 2, 4, 10, 0, 2, 7, 2, 5, 11, 4, 2, 3, 7, 4, 11, 7, 2, 12, 2, 0, 11, 4, 14, 11, 2, 4, 11, 5, 2, 14, 2, 8, 25, 4, 2, 3, 9, 9, 11, 8, 2, 16, 17, 7, 11, 4, 2, 16, 2, 4, 27, 0, 17, 18, 2, 8, 11, 16
Offset: 0
n=15 is the sum of three partitions of n with two or more consecutive positive integers: 15 = 1 + 2 + 3 + 4 + 5, 15 = 4 + 5 + 6, 15 = 7 + 8.
The sum of the lengths of these partitions is a(15) = 5 + 3 + 2 = 10.
On the other hand a(8) = 0 because there are no partitions of two or more consecutive integers adding up to 8.
A204218
G.f. satisfies: A(x) = 1 + Sum_{n>=1} n * (x*A(x))^(n*(n+1)/2) / (1 - x^n*A(x)^n).
Original entry on oeis.org
1, 1, 2, 7, 22, 74, 271, 1012, 3858, 15046, 59579, 238825, 967873, 3958517, 16316594, 67716000, 282719162, 1186633647, 5004102122, 21192022233, 90089538788, 384305738731, 1644544501988, 7057705570877, 30368821119351, 130993073168419, 566297366630412, 2453269044761359
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 74*x^5 + 271*x^6 + 1012*x^7 +...
where A(x) = 1 + x*A(x)/(1 - x*A(x)) + 2*x^3*A(x)^3/(1 - x^2*A(x)^2) + 3*x^6*A(x)^6/(1 - x^3*A(x)^3) + 4*x^10*A(x)^10/(1 - x^4*A(x)^4) +...
Also, by a Ramanujan identity:
(A(x) - 1)*Theta4(x*A(x))^2 = x*A(x)*(1 - x*A(x))/(1 + x*A(x))^2 - 2*x^3*A(x)^3*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2 + 3*x^6*A(x)^6*(1 - x^3*A(x)^3)/(1 + x^3*A(x)^3)^2 - 4*x^10*A(x)^10*(1 - x^4*A(x)^4)/(1 + x^4*A(x)^4)^2 +...
-
a[n_] := (For[A = 1+x; i = 1, i <= n, i++, A = 1+Sum[m*(x*A)^(m*(m+1)/2)/(1 - (x*A + x*O[x]^n)^m), {m, 1, n}]]; Coefficient[A, x, n]); Table[an = a[n]; Print[an]; an, {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017, translated from PARI *)
-
{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,m*(x*A)^(m*(m+1)/2)/(1-(x*A +x*O(x^n))^m)));polcoeff(A,n)}
-
{a(n)=local(A=1+x,Theta4=1+2*sum(m=1,sqrtint(n+1),(-x)^(m^2))); for(i=1,n,A=1+1/subst(Theta4^2,x,x*A+x*O(x^n))*sum(m=1,sqrtint(2*n+1),(-1)^(m-1)*m*(x*A)^(m*(m+1)/2)*(1-(x*A)^m)/(1+(x*A)^m+x*O(x^n))^2));polcoeff(A,n)}
A319895
a(n) is the number of partitions of n into consecutive parts, plus the total number of parts in those partitions.
Original entry on oeis.org
2, 2, 5, 2, 5, 6, 5, 2, 9, 7, 5, 6, 5, 7, 15, 2, 5, 11, 5, 8, 16, 7, 5, 6, 11, 7, 16, 10, 5, 17, 5, 2, 16, 7, 19, 15, 5, 7, 16, 8, 5, 19, 5, 11, 32, 7, 5, 6, 13, 13, 16, 11, 5, 21, 22, 10, 16, 7, 5, 21, 5, 7, 34, 2, 22, 23, 5, 11, 16, 21, 5, 16, 5, 7, 33, 11, 25, 24, 5, 8, 26, 7, 5, 23, 22, 7, 16, 14, 5
Offset: 1
Illustration of a diagram of partitions into consecutive parts (first 28 rows):
. _
. _|1
. _|2 _
. _|3 |2
. _|4 _|1
. _|5 |3 _
. _|6 _|2|3
. _|7 |4 |2
. _|8 _|3 _|1
. _|9 |5 |4 _
. _|10 _|4 |3|4
. _|11 |6 _|2|3
. _|12 _|5 |5 |2
. _|13 |7 |4 _|1
. _|14 _|6 _|3|5 _
. _|15 |8 |6 |4|5
. _|16 _|7 |5 |3|4
. _|17 |9 _|4 _|2|3
. _|18 _|8 |7 |6 |2
. _|19 |10 |6 |5 _|1
. _|20 _|9 _|5 |4|6 _
. _|21 |11 |8 _|3|5|6
. _|22 _|10 |7 |7 |4|5
. _|23 |12 _|6 |6 |3|4
. _|24 _|11 |9 |5 _|2|3
. _|25 |13 |8 _|4|7 |2
. _|26 _|12 _|7 |8 |6 _|1
. _|27 |14 |10 |7 |5|7 _
. |28 |13 |9 |6 |4|6|7
...
For n = 21 we have that there are four partitions of 21 into consecutive parts, they are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1]. The total number of parts is 1 + 2 + 3 + 6 = 12. Therefore the number of partitions plus the total number of parts is 4 + 12 = 16, so a(21) = 16.
On the other hand, in the above diagram there are four pairs of orthogonal line segments whose horizontal upper part are located on the 21st row, as shown below:
. _ _ _ _
. |21 |11 |8 |6
. |10 |7 |5
. |6 |4
. |3
. |2
. |1
.
The four horizontal line segments have length 1, and the vertical line segments have lengths 1, 2, 3, 6 respectively. Therefore the total length of the line segments is 1 + 1 + 1 + 1 + 1 + 2 + 3 + 6 = 16, so a(21) = 16.
For tables of partitions into consecutive parts see
A286000 and
A286001.
Cf.
A000079,
A001227,
A065091,
A204217,
A237048,
A237593,
A285898,
A285899,
A285900,
A285900,
A285901,
A285902,
A288529,
A288772,
A288773,
A288774,
A299765.
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