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

1,1

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).

Links

Table of n, a(n) for n=1..60.

Formula

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

Example

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).

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], 0<Total[1/Select[Divisors[#], EvenQ]]<1&] (* Harvey P. Dale, Jan 10 2024 *)

Crossrefs

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

Sequence in context: A006586 A260391 A022292 * A087370 A364158 A138929

Adjacent sequences: A225238 A225239 A225240 * A225242 A225243 A225244

Keyword

nonn

Author

Michel Lagneau, Jul 25 2013

Extensions

Definition corrected by Harvey P. Dale, Jan 10 2024

Status

approved