A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).

12, 56, 992, 16256, 67100672, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432

Offset: 1

Author

Omar E. Pol, Apr 22 2008

Keywords

• nonn

Comments

Also, twice perfect numbers, if there are no odd perfect numbers.

If there are no odd perfect numbers, essentially the same as A065125. - R. J. Mathar, May 23 2008

The sum of reciprocals of even divisors of a(n) equals 1. Proof: Let n = (2^m - 1)*2^m where 2^m - 1 is a Mersenne prime. The sum of reciprocals of even divisors of n is s1 + s2 where: s1 = 1/2 + 1/4 + ... + 1/2^m = (2^m - 1)/2^m and s2 = s1/(2^m - 1) => s1 + s2 = 1. - Michel Lagneau, Jul 17 2013

Examples

			a(3) = 992 because the third Mersenne prime A000668(3) is 31 and 31*(31+1) = 31*32 = 992.
a(3) = 992 because the sum of the divisors of the third perfect number is 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 992. - _Omar E. Pol_, Dec 05 2016
From _Omar E. Pol_, Aug 13 2021: (Start)
Illustration of initial terms in which a(n) is represented as the sum of the divisors of the n-th even perfect number P(n).
-------------------------------------------------------------------------
  n  P(n) a(n)  Diagram:   1                                           2
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                           _                                           _
                          | |                                         | |
                          | |                                         | |
                       _ _| |                                         | |
                      |    _|                                         | |
                 _ _ _|  _|                                           | |
  1    6   12   |_ _ _ _|                                             | |
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                                                             _ _ _ _ _| |
                                                            |  _ _ _ _ _|
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                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|
                                             _ _ _| |
                                            |  _ _ _|
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                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  2   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) equals the area (also the number of cells) in the n-th diagram.
For n = 3, P(3) = 496 and a(3) = 992, 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]. For a definition of these numbers related to partitions into consecutive parts see A237591. (End)
		

Links

• Walter A. Kehowski, Power-spectral Numbers, ResearchGate (2024); also available at vixra.org.
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Crossrefs

Cf. A000203, A000396, A000668, A001065, A065125, A139257, A237591, A237593, A245092.

Programs

Mathematica

DeleteCases[2 Map[(# (# + 1))/2 &, Select[2^Range[100] - 1, PrimeQ]], k_ /; OddQ@ k] (* Michael De Vlieger, Dec 05 2016, after Harvey P. Dale at A000396 *)

Formula

a(n) = A000668(n)*(A000668(n)+1).

a(n) = 2*A000396(n), if there are no odd perfect numbers.

a(n) = A000203(A000396(n)) = A001065(A000396(n)) + A000396(n), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 04 2016

Extensions

More terms from Omar E. Pol, Jun 07 2012