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A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n). 11
12, 56, 992, 16256, 67100672, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432 (list; graph; refs; listen; history; text; internal format)
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

Numbers k such that k = sigma(j), where j is the greatest aliquot part of k. - Paolo P. Lava, May 16 2017

LINKS

Table of n, a(n) for n=1..10.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

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

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

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

Cf. A000203, A000396, A000668, A001065, A065125, A139257.

Sequence in context: A095724 A225880 A224832 * A166997 A204674 A123983

Adjacent sequences:  A139253 A139254 A139255 * A139257 A139258 A139259

KEYWORD

nonn

AUTHOR

Omar E. Pol, Apr 22 2008

EXTENSIONS

More terms from Omar E. Pol, Jun 07 2012

STATUS

approved

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Last modified July 25 04:48 EDT 2017. Contains 289779 sequences.