OFFSET
1,1
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
LINKS
Walter A. Kehowski, Power-spectral Numbers, ResearchGate (2024); also available at vixra.org.
FORMULA
EXAMPLE
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).
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n P(n) a(n) Diagram: 1 2
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2 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
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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)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Apr 22 2008
EXTENSIONS
More terms from Omar E. Pol, Jun 07 2012
STATUS
approved