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

12, 56, 992, 16256, 60480, 65520, 4357080, 47139840, 67100672, 91065600, 285981696, 2758909440, 17179738112, 87722956800, 132867440640, 274877382656, 306007080960, 806062473216, 1409150457792, 363485766938112, 12177456042320640, 29884246553283840, 40316371715063808

Offset: 1

Author

Michel Lagneau, Jul 21 2013

Keywords

• nonn

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.

Examples

			12 is in the sequence because the divisors are {1, 2, 3, 4, 6, 12} and 1/2 + 1/4 + 1/6 + 1/12 = 1 is an 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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1013

Crossrefs

Cf. A000668, A139256, A194771, A325637.

Programs

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:

Formula

a(n) = 2*A325637(n). - Amiram Eldar, Jun 26 2024

Extensions

a(17)-a(23) from Amiram Eldar, Jun 26 2024