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A224832
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Numbers n such that the sum of reciprocals of even divisors of n is an integer.
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2
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12, 56, 992, 16256, 60480, 65520, 4357080, 47139840, 67100672, 91065600, 285981696, 2758909440, 17179738112, 87722956800, 132867440640, 274877382656
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OFFSET
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1,1
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COMMENTS
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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 non-twice 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 non-twice even perfect numbers, the sum of reciprocals of even divisors equals 2.
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LINKS
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EXAMPLE
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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 = 2^13-1 and 8191*(8191+1) = 8191*8192 = 67100672.
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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