%I #10 Jan 18 2016 10:26:24
%S 2,3,5,8,11,19,38,87,106,193,386,1264,1457,2721,5442,8163,23225,25946,
%T 49171,98342,147513,517656,566827,1084483,2168966,3253449,13580623,
%U 14665106,28245729,56491458,84737187,112982916,141228645
%N Integers in the interval [e*k - 1/k, e*k + 1/k] for some k >0 , where e = 2.71828... is Euler's number.
%C Conjecture: the sequence is infinite.
%C See the reference for a similar problem with Fibonacci numbers.
%C The corresponding integers k are 1, 2, 3, 4, 7, 14, 32, 39, 71, 142, 465, ...(see A265742)
%C For k > 1, the interval [e*k - 1/k, e*k + 1/k] contains exactly one integer.
%C We observe two properties:
%C (1) a(n) = m*a(n-m+1) for some n, m=2,3,4 and 5
%C Examples:
%C m = 2 => a(7)=2*a(6), a(11)=2*a(10), a(15)=2*a(14), a(20)=2*a(19), a(25)=2*a(24), a(30)=2*a(29), ...
%C m = 3 => a(16)=3*a(14), a(21)=3*a(19), a(26)=3*a(24), a(31)=3*a(29), ...
%C m = 4 => a(4)=4*a(1), a(32)=4*a(29), ...
%C m = 5 => a(33)=5*a(29), ...
%C (2) a(n+2) = a(n) + a(n+1) for n = 1, 3, 7, 11, 13, 16, 18, 21, 23, 26, 28, ...
%H Takao Komatsu, <a href="http://www.fq.math.ca/Scanned/41-1/komatsu.pdf">The interval associated with a Fibonacci number</a>, The Fibonacci Quarterly, Volume 41, Number 1, February 2003.
%e For k=1, there exist two integers, a(1)=2 and a(2)=3, in the interval [1*e -1/1, 1*e + 1/1] = [1.71828..., 3.71828...];
%e for k=2, the number a(3)=5 belongs to the interval [2*e-1/2, 2*e+1/2] = [4.936564..., 5.936564...];
%e for k=3, the number a(4)= 8 belongs to the interval [3*e-1/3, 3*e+1/3] = [7.821512..., 8.488179...].
%p *** the program gives the interval [a,b],a(n) and k ***
%p nn:=10^9:
%p e:=exp(1):
%p for n from 1 to nn do:
%p x1:=evalhf(e*n-1/n):y1:=evalhf(e*n+1/n):
%p x:=floor(x1):y:=floor(y1):
%p for j from x+1 to y do:
%p printf("%g %g %d %d\n",x1,y1,j,n):
%p od:
%p od:
%Y Cf. A001113, A265742.
%K nonn
%O 1,1
%A _Michel Lagneau_, Dec 15 2015