

A224832


Numbers n such that the sum of reciprocals of even divisors of n is an integer.


2



12, 56, 992, 16256, 60480, 65520, 4357080, 47139840, 67100672, 91065600, 285981696, 2758909440, 17179738112, 87722956800, 132867440640, 274877382656
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OFFSET

1,1


COMMENTS

This sequence is a subsequence of A194771. The sequence A139256 (twice even perfect numbers) is a subsequence and the sum of the reciprocals of even divisors equals 1 (see the proof in this sequence). But, for the nontwice even perfect numbers of this sequence, for example a(5) = 60480, a(6) = 65520, a(7) = 4357080 so the sum of the reciprocals of even divisors equals 2.
Conjecture: if a(n) is a nontwice even perfect numbers, the sum of reciprocals of even divisors equals 2.


LINKS

Table of n, a(n) for n=1..16.


EXAMPLE

12 is in the sequence because de divisors are {1, 2, 3, 4, 6, 12} and 1/2 + 1/4 + 1/6 + 1/12 = 1 is integer.
67100672 is in the sequence because a(8)=A139256(5), the 5th Mersenne prime A000668(5) is 8191 = 2^131 and 8191*(8191+1) = 8191*8192 = 67100672.


MAPLE

with(numtheory):for n from 2 to 200000 do:x:=divisors(n):n1:=nops(x): s:=0:for i from 1 to n1 do: if irem(x[i], 2)=0 then s:=s+1/x[i]:else fi:od: if s>0 and s=floor(s) then print(n):else fi:od:


CROSSREFS

Cf. A139256, A194771, A000668.
Sequence in context: A027147 A095724 A225880 * A139256 A166997 A204674
Adjacent sequences: A224829 A224830 A224831 * A224833 A224834 A224835


KEYWORD

nonn,hard


AUTHOR

Michel Lagneau, Jul 21 2013


STATUS

approved



