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Numbers n such that the sum of the reciprocals of the even divisors of n is greater than zero and less than one.
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%I #12 Jan 10 2024 19:15:42

%S 2,4,6,8,10,14,16,18,20,22,26,28,30,32,34,38,42,44,46,50,52,54,58,62,

%T 64,66,68,70,74,76,78,82,86,88,90,92,94,98,100,102,104,106,110,114,

%U 116,118,122,124,126,128,130,134,136,138,142,146,148,150,152,154

%N Numbers n such that the sum of the reciprocals of the even divisors of n is greater than zero and less than one.

%C Numbers n such that the sum of reciprocals of even divisors of n equals m/n for some integer m where the fraction m/n < 1.

%C The corresponding numerators m are given by the sequence A204823(n) = {1, 3, 4, 7, 6, 8, 15, 13, 18, 12, 14, 24,...} (Sum of divisors (A000203) of deficient numbers (A005100).

%F a(n) = 2*A005100(n) where A005100 are deficient numbers: numbers n such that sigma(n) < 2n.

%e 8 is in the sequence because the even divisors of 8 are 2, 4, 8 and 1/2 + 1/4 + 1/8 = 7/8 = A204823(4)/a(4).

%p ***program 1 where sum of reciprocals even divisors < 1***

%p with(numtheory):for n from 2 by 2 to 500 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<1 then printf(`%d, `, n):else fi:od:

%p ***program 2 where sum of reciprocals even divisors = m/n***

%p with(numtheory):for n from 2 to 500 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: for m from 1 to n-1 do: if s=m/n then printf(`%d, `,n):else fi:od:od:

%t Select[Range[200],0<Total[1/Select[Divisors[#],EvenQ]]<1&] (* _Harvey P. Dale_, Jan 10 2024 *)

%Y Cf. A000203, A005100, A204823, A224832, A224907.

%K nonn

%O 1,1

%A _Michel Lagneau_, Jul 25 2013

%E Definition corrected by _Harvey P. Dale_, Jan 10 2024