A224907 Numbers n such that the sum of reciprocals of even divisors of n > 1.

24, 36, 40, 48, 60, 72, 80, 84, 96, 108, 112, 120, 132, 140, 144, 156, 160, 168, 176, 180, 192, 200, 204, 208, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 320, 324, 336, 348, 352, 360, 372, 384, 392, 396, 400, 408, 416, 420, 432, 440, 444, 448, 456

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 A204822(n) = {28, 39, 42, 60, 72, 91, 90, 96,...} (Sum of divisors (A000203) of abundant numbers (A005101)).

Examples

			40 is in the sequence because the even divisors of 40 are 2, 4, 8, 10, 20, 40 and 1/2 + 1/4 + 1/8 + 1/10 + 1/20 + 1/40 = 42/40 = A204823(3)/a(3), and 42/40 > 1.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203, A005101, A204822, A224832, A225241.

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 n+1 to 2*n do: if s=m/n then printf(`%d, `,n):else fi:od:od:

Mathematica

Select[Range[500],Total[1/Select[Divisors[#],EvenQ]]>1&] (* Harvey P. Dale, Aug 15 2015 *)

Formula

a(n) = 2*A005101(n).