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%I #54 Jun 16 2023 05:31:25
%S 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,
%T 1,2,1,2,1,1,1,1,1,1,1,4065,1355,678,407,271,194,146,113,91,75,62,52,
%U 45,40,34,30,27,25,22,19,19,16,15,14,13,12,12,10,10,9,9,8,8,7
%N Irregular triangle read by rows in which row n is the "n-th even perfect number" row of A237591, n >= 1.
%C 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.
%C 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).
%C Therefore we can see a geometric pattern of the distribution of the even perfect numbers in the stepped pyramid described in A245092.
%C For the definition of "width" see A249351.
%C 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.
%C 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.
%H Michel Marcus, <a href="/A346876/b346876.txt">Table of n, a(n) for n = 1..8359</a> (rows 1..5).
%e Triangle begins:
%e 4, 1, 1;
%e 15, 5, 3, 2, 1, 1,1;
%e 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;
%e ...
%e Illustration of initial terms:
%e Column P gives the even perfect numbers (A000396 assuming there are no odd perfect numbers).
%e 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.
%e -------------------------------------------------------------------------
%e n P S Diagram: 1 2
%e -------------------------------------------------------------------------
%e _ _
%e | | | |
%e | | | |
%e _ _| | | |
%e | _| | |
%e _ _ _| _| | |
%e 1 6 12 |_ _ _ _| 1 | |
%e 4 1 | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e _ _ _ _ _| |
%e | _ _ _ _ _|
%e | |
%e _ _| |
%e _ _| _ _|
%e | _|
%e _| _|
%e | _|1 1
%e _ _ _| | 1
%e | _ _ _|2
%e | | 3
%e | |
%e | |5
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e 2 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e 15
%e .
%e 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].
%o (PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
%o row(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); } \\ A237591
%o tabf(nn) = {for (n=1, nn, my(p=prime(n)); if (isprime(2^n-1), print(row(2^(n-1)*(2^n-1)));););}
%o tabf(7) \\ _Michel Marcus_, Aug 31 2021
%Y Row sums give A000396.
%Y Row lengths give A000668.
%Y For the characteristic shape of sigma(A000040(n)) see A346871.
%Y For the characteristic shape of sigma(A000079(n)) see A346872.
%Y For the characteristic shape of sigma(A000217(n)) see A346873.
%Y For the visualization of Mersenne numbers A000225 see A346874.
%Y For the characteristic shape of sigma(A000384(n)) see A346875.
%Y For the characteristic shape of sigma(A008588(n)) see A224613.
%Y Cf. A000203, A139256, A237591, A237593, A245092, A249351, A262626.
%K nonn,tabf
%O 1,1
%A _Omar E. Pol_, Aug 06 2021
%E More terms from _Michel Marcus_, Aug 31 2021
%E Name edited by _Michel Marcus_, Jun 16 2023