OFFSET
1,1
COMMENTS
The characteristic shape of the symmetric representation of sigma(A000396(n)) consists in that the diagram has only one region (or part) and that region has whidth 1 except in the main diagonal where the width is 2.
So knowing this characteristic shape we can know if a number is an even perfect number (or not) just by looking at the diagram, even ignoring the concept of even perfect number (see the examples).
Therefore we can see a geometric pattern of the distribution of the even perfect numbers in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000396(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000396(n) assuming there are no odd perfect numbers.
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th even perfect number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th perfect number into exactly k + 1 consecutive parts.
LINKS
Michel Marcus, Table of n, a(n) for n = 1..8359 (rows 1..5).
EXAMPLE
Triangle begins:
4, 1, 1;
15, 5, 3, 2, 1, 1,1;
249,83,42,25,17,13,9,7,6,5,5,3,4,2,3,2,2,2,2,2,1,2,1,2,1,1,1,1,1,1,1;
...
Illustration of initial terms:
Column P gives the even perfect numbers (A000396 assuming there are no odd perfect numbers).
Column S gives A139256, the sum of the divisors of the even perfect numbers equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
n P S Diagram: 1 2
-------------------------------------------------------------------------
_ _
| | | |
| | | |
_ _| | | |
| _| | |
_ _ _| _| | |
1 6 12 |_ _ _ _| 1 | |
4 1 | |
| |
| |
| |
| |
| |
| |
_ _ _ _ _| |
| _ _ _ _ _|
| |
_ _| |
_ _| _ _|
| _|
_| _|
| _|1 1
_ _ _| | 1
| _ _ _|2
| | 3
| |
| |5
_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
2 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
15
.
For n = 3, P = 496, the diagram is too large to include here. To draw that diagram note that the lengths of the line segments of the smallest Dyck path are [248, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 248] and the lengths of the line segments of the largest Dyck path are [249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 249].
PROG
(PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
row(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); } \\ A237591
tabf(nn) = {for (n=1, nn, my(p=prime(n)); if (isprime(2^n-1), print(row(2^(n-1)*(2^n-1))); ); ); }
tabf(7) \\ Michel Marcus, Aug 31 2021
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 06 2021
EXTENSIONS
More terms from Michel Marcus, Aug 31 2021
Name edited by Michel Marcus, Jun 16 2023
STATUS
approved