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A265739 Numbers k such that there exists at least one integer in the interval [Pi*k - 1/k, Pi*k + 1/k]. 1
1, 2, 6, 7, 14, 21, 28, 106, 113, 226, 339, 452, 565, 678, 791, 904, 1017, 1130, 1243, 1356, 1469, 1582, 1695, 1808, 1921, 33102, 33215, 66317, 99532, 165849, 265381, 364913, 729826, 1360120, 1725033, 3450066, 5175099, 25510582, 27235615 (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.

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

The corresponding integers in the interval [Pi*k - 1/k, Pi*k + 1/k] are 3, 4, 6, 19, 22, 44, 66, 88, ... (see A265735).

LINKS

Table of n, a(n) for n=1..39.

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, 3 and 4, in the interval [1*Pi - 1/1, 1*Pi + 1/1] = [2.14159..., 4.14159...];

for k=2, the number 6 is in the interval [2*Pi - 1/2, 2*Pi + 1/2] = [5.783185..., 6.783185...].

for k=6, the number 19 is in the interval [6*Pi - 1/6, 6*Pi + 1/6] = [18.682889..., 19.016223...].

MAPLE

# program gives the interval [a, b], the first integer in [a, b] and n

nn:=10^9:

for n from 1 to nn do:

x1:=evalhf(Pi*n-1/n):y1:=evalhf(Pi*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. A000796, A265735.

Sequence in context: A087376 A199974 A176279 * A281167 A210660 A226965

Adjacent sequences:  A265736 A265737 A265738 * A265740 A265741 A265742

KEYWORD

nonn

AUTHOR

Michel Lagneau, Dec 15 2015

STATUS

approved

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Last modified May 28 14:47 EDT 2020. Contains 334684 sequences. (Running on oeis4.)