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A224832 Numbers n such that the sum of reciprocals of even divisors of n is an integer. 2

%I #7 Jul 22 2013 00:57:03

%S 12,56,992,16256,60480,65520,4357080,47139840,67100672,91065600,

%T 285981696,2758909440,17179738112,87722956800,132867440640,

%U 274877382656

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

%C 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.

%C Conjecture: if a(n) is a non-twice even perfect numbers, the sum of reciprocals of even divisors equals 2.

%e 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.

%e 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.

%p 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:

%Y Cf. A139256, A194771, A000668.

%K nonn,hard

%O 1,1

%A _Michel Lagneau_, Jul 21 2013

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Last modified April 25 07:07 EDT 2024. Contains 371964 sequences. (Running on oeis4.)