Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #63 Sep 29 2024 23:46:11
%S 12,56,992,16256,67100672,17179738112,274877382656,
%T 4611686016279904256,5316911983139663489309385231907684352,
%U 383123885216472214589586756168607276261994643096338432
%N Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).
%C Also, twice perfect numbers, if there are no odd perfect numbers.
%C If there are no odd perfect numbers, essentially the same as A065125. - _R. J. Mathar_, May 23 2008
%C 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
%H Walter A. Kehowski, <a href="https://www.researchgate.net/publication/383862795_Power-spectral_Numbers">Power-spectral Numbers</a>, ResearchGate (2024); also available at <a href="https://www.vixra.org/pdf/2409.0031v1.pdf">vixra.org</a>.
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F a(n) = A000668(n)*(A000668(n)+1).
%F a(n) = 2*A000396(n), if there are no odd perfect numbers.
%F a(n) = A000203(A000396(n)) = A001065(A000396(n)) + A000396(n), assuming there are no odd perfect numbers. - _Omar E. Pol_, Dec 04 2016
%e a(3) = 992 because the third Mersenne prime A000668(3) is 31 and 31*(31+1) = 31*32 = 992.
%e 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
%e From _Omar E. Pol_, Aug 13 2021: (Start)
%e 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).
%e -------------------------------------------------------------------------
%e n P(n) a(n) Diagram: 1 2
%e -------------------------------------------------------------------------
%e _ _
%e | | | |
%e | | | |
%e _ _| | | |
%e | _| | |
%e _ _ _| _| | |
%e 1 6 12 |_ _ _ _| | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e _ _ _ _ _| |
%e | _ _ _ _ _|
%e | |
%e _ _| |
%e _ _| _ _|
%e | _|
%e _| _|
%e | _|
%e _ _ _| |
%e | _ _ _|
%e | |
%e | |
%e | |
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e 2 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e .
%e a(n) equals the area (also the number of cells) in the n-th diagram.
%e 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)
%t 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 *)
%Y Cf. A000203, A000396, A000668, A001065, A065125, A139257, A237591, A237593, A245092.
%K nonn
%O 1,1
%A _Omar E. Pol_, Apr 22 2008
%E More terms from _Omar E. Pol_, Jun 07 2012