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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].

%I A265739 #25 Mar 29 2025 10:45:15
%S A265739 1,2,6,7,14,21,28,106,113,226,339,452,565,678,791,904,1017,1130,1243,
%T A265739 1356,1469,1582,1695,1808,1921,33102,33215,66317,99532,165849,265381,
%U A265739 364913,729826,1360120,1725033,3450066,5175099,25510582,27235615
%N A265739 Numbers k such that there exists at least one integer in the interval [Pi*k - 1/k, Pi*k + 1/k].
%C A265739 Conjecture: the sequence is infinite.
%C A265739 See the reference for a similar problem with Fibonacci numbers.
%C A265739 For k > 1, the interval [Pi*k - 1/k, Pi*k + 1/k] contains exactly one integer.
%C A265739 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).
%C A265739 The sequence is infinite by Dirichlet's approximation theorem. In other words, the irrationality measure of Pi is at least 2 so this sequence is infinite. - _Charles R Greathouse IV_, Nov 07 2022
%H A265739 Takao Komatsu, <a href="https://web.archive.org/web/2024*/https://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.
%H A265739 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>
%e A265739 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...];
%e A265739 for k=2, the number 6 is in the interval [2*Pi - 1/2, 2*Pi + 1/2] = [5.783185..., 6.783185...].
%e A265739 for k=6, the number 19 is in the interval [6*Pi - 1/6, 6*Pi + 1/6] = [18.682889..., 19.016223...].
%p A265739 # program gives the interval [a,b], the first integer in [a,b] and n
%p A265739 nn:=10^9:
%p A265739 for n from 1 to nn do:
%p A265739 x1:=evalhf(Pi*n-1/n):y1:=evalhf(Pi*n+1/n):
%p A265739 x:=floor(x1):y:=floor(y1):
%p A265739 for j from x+1 to y do:
%p A265739 printf("%g %g %d %d\n",x1,y1,j,n):
%p A265739 od:
%p A265739 od:
%Y A265739 Cf. A000796, A265735. Contains A002486 (without the first two terms) as a subsequence.
%K A265739 nonn
%O A265739 1,2
%A A265739 _Michel Lagneau_, Dec 15 2015