A224832 Numbers n such that the sum of reciprocals of even divisors of n is an integer.

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

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 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.

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 = 2^13-1 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