A265741 Integers in the interval [e*k - 1/k, e*k + 1/k] for some k >0 , where e = 2.71828... is Euler's number.

2, 3, 5, 8, 11, 19, 38, 87, 106, 193, 386, 1264, 1457, 2721, 5442, 8163, 23225, 25946, 49171, 98342, 147513, 517656, 566827, 1084483, 2168966, 3253449, 13580623, 14665106, 28245729, 56491458, 84737187, 112982916, 141228645

Offset

1,1

Comments

Conjecture: the sequence is infinite.

See the reference for a similar problem with Fibonacci numbers.

The corresponding integers k are 1, 2, 3, 4, 7, 14, 32, 39, 71, 142, 465, ...(see A265742)

For k > 1, the interval [e*k - 1/k, e*k + 1/k] contains exactly one integer.

We observe two properties:

(1) a(n) = m*a(n-m+1) for some n, m=2,3,4 and 5

Examples:

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), ...

m = 3 => a(16)=3*a(14), a(21)=3*a(19), a(26)=3*a(24), a(31)=3*a(29), ...

m = 4 => a(4)=4*a(1), a(32)=4*a(29), ...

m = 5 => a(33)=5*a(29), ...

(2) a(n+2) = a(n) + a(n+1) for n = 1, 3, 7, 11, 13, 16, 18, 21, 23, 26, 28, ...

Links

Table of n, a(n) for n=1..33.

Takao Komatsu, The interval associated with a Fibonacci number, The Fibonacci Quarterly, Volume 41, Number 1, February 2003.

Example

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...];
for k=2, the number a(3)=5 belongs to the interval [2*e-1/2, 2*e+1/2] = [4.936564..., 5.936564...];
for k=3, the number a(4)= 8 belongs to the interval [3*e-1/3, 3*e+1/3] = [7.821512..., 8.488179...].

Maple

*** the program gives the interval [a, b], a(n) and k ***
nn:=10^9:
e:=exp(1):
for n from 1 to nn do:
x1:=evalhf(e*n-1/n):y1:=evalhf(e*n+1/n):
x:=floor(x1):y:=floor(y1):
for j from x+1 to y do:
printf("%g %g %d %d\n", x1, y1, j, n):
od:
od:

Crossrefs

Cf. A001113, A265742.

Sequence in context: A119014 A006258 A177967 * A254351 A346116 A262841

Adjacent sequences: A265738 A265739 A265740 * A265742 A265743 A265744

Keyword

nonn

Author

Michel Lagneau, Dec 15 2015

Status

approved