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A265742 Numbers k such that there exists at least an integer in the interval [e*k - 1/k, e*k + 1/k] where e = 2.71828... is Euler's number. 1
1, 2, 3, 4, 7, 14, 32, 39, 71, 142, 465, 536, 1001, 2002, 3003, 8544, 9545, 18089, 36178, 54267, 190435, 208524, 398959, 797918, 1196877, 4996032, 5394991, 10391023, 20782046, 31173069, 41564092, 51955115 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: the sequence is infinite.

See the reference for a similar problem with Fibonacci numbers.

The corresponding integers in each interval [e*k - 1/k, e*k + 1/k] are 2, 3, 5, 8, 11, 19, 38, 87, 106, 193, 386, 1264, ...(see A265741).

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), ...

(ii) 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..32.

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

EXAMPLE

a(1) = 1 because there exist two integers (2 and 3) in the interval [1*e -1/1, 1*e + 1/1] = [1.71828..., 3.71828...];

a(2) = 2 because the number 5 belongs to the interval [2*e-1/2, 2*e+1/2] = [4.936564..., 5.936564...];

a(3) = 3 because the number 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], the integer(s) between [a, b] 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 %g %g\n", x1, y1, j, n):

od:

od:

CROSSREFS

Cf. A001113, A265741.

Sequence in context: A049795 A014251 A290992 * A098010 A088533 A091155

Adjacent sequences:  A265739 A265740 A265741 * A265743 A265744 A265745

KEYWORD

nonn,more

AUTHOR

Michel Lagneau, Dec 15 2015

STATUS

approved

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Last modified July 20 22:54 EDT 2019. Contains 325189 sequences. (Running on oeis4.)