A225241 Numbers n such that the sum of the reciprocals of the even divisors of n is greater than zero and less than one.

2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 26, 28, 30, 32, 34, 38, 42, 44, 46, 50, 52, 54, 58, 62, 64, 66, 68, 70, 74, 76, 78, 82, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 110, 114, 116, 118, 122, 124, 126, 128, 130, 134, 136, 138, 142, 146, 148, 150, 152, 154

Offset: 1

Author

Michel Lagneau, Jul 25 2013

Keywords

• nonn

Comments

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.

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)).

Examples

			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).
		

Crossrefs

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

Programs

Maple

***program 1 where sum of reciprocals even divisors < 1***
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:
***program 2 where sum of reciprocals even divisors = m/n***
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:

Mathematica

Select[Range[200],0Harvey P. Dale, Jan 10 2024 *)

Formula

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

Extensions

Definition corrected by Harvey P. Dale, Jan 10 2024