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

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%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