cp's OEIS Frontend

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

%I A224832 #19 Jan 15 2025 01:44:29
%S A224832 12,56,992,16256,60480,65520,4357080,47139840,67100672,91065600,
%T A224832 285981696,2758909440,17179738112,87722956800,132867440640,
%U A224832 274877382656,306007080960,806062473216,1409150457792,363485766938112,12177456042320640,29884246553283840,40316371715063808
%N A224832 Numbers k such that the sum of reciprocals of even divisors of k is an integer.
%C A224832 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 A224832 Conjecture: if a(n) is a non-twice even perfect numbers, the sum of reciprocals of even divisors equals 2.
%H A224832 Amiram Eldar, <a href="/A224832/b224832.txt">Table of n, a(n) for n = 1..1013</a>
%F A224832 a(n) = 2*A325637(n). - _Amiram Eldar_, Jun 26 2024
%e A224832 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.
%e A224832 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 A224832 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 A224832 Cf. A000668, A139256, A194771, A325637.
%K A224832 nonn
%O A224832 1,1
%A A224832 _Michel Lagneau_, Jul 21 2013
%E A224832 a(17)-a(23) from _Amiram Eldar_, Jun 26 2024