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A265735 Integers in the interval [Pi*k - 1/k, Pi*k + 1/k] for some k > 0.

%I A265735 #35 Mar 29 2025 10:45:10
%S A265735 3,4,6,19,22,44,66,88,333,355,710,1065,1420,1775,2130,2485,2840,3195,
%T A265735 3550,3905,4260,4615,4970,5325,5680,6035,103993,104348,208341,312689,
%U A265735 521030,833719,1146408,2292816,4272943,5419351,10838702,16258053,80143857,85563208
%N A265735 Integers in the interval [Pi*k - 1/k, Pi*k + 1/k] for some k > 0.
%C A265735 Conjecture: the sequence is infinite.
%C A265735 See the reference for a similar problem with Fibonacci numbers.
%C A265735 For k > 1, the interval [Pi*k - 1/k, Pi*k + 1/k] contains exactly one integer.
%C A265735 The corresponding integers k are 1, 2, 6, 7, 14, 21, 28,...(see A265739).
%C A265735 We observe two properties:
%C A265735 (1) a(n) = m*a(n-m+1) for some n, m=2,3,4.
%C A265735 Examples:
%C A265735 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 A265735 m = 3 => a(16)=3*a(14), a(21)=3*a(19), a(26)=3*a(24), a(31)=3*a(29),...
%C A265735 m = 4 => a(4)=4*a(1), a(32)=4*a(29), ...
%C A265735 But, for m=5, the formula (1) is not valid. We find a(13)=5*a(9), a(18)=5*a(10), a(23)=5*a(11), ...
%C A265735 (2) a(n+2) = a(n) + a(n+1) for n = 4, 9, 26, 27, 28, 29, 35, ...
%C A265735 For k > 1, the integer satisfying the definition is such that ceiling(Pi*k - 1/k) = floor(Pi*k + 1/k). - _Stefano Spezia_, Apr 26 2023
%H A265735 Stefano Spezia, <a href="/A265735/b265735.txt">Table of n, a(n) for n = 1..52</a>
%H A265735 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 A265735 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>
%e A265735 For k=1 there exists two integers a(1)=3 and a(2)=4 in the interval [1*Pi -1/1, 1*Pi + 1/1] = [2.14159...,4.14159...];
%e A265735 for k=2, the number a(3)=6 is in the interval [2*Pi-1/2, 2*Pi+1/2] = [5.783185..., 6.783185...];
%e A265735 for k=6, the number a(4)= 19 is in the interval [6*Pi-1/6, 6*Pi+1/6] = [18.682889..., 19.016223...].
%p A265735 *** the program gives the interval [a,b],a(n) and k ***
%p A265735 nn:=10^9:
%p A265735 for n from 1 to nn do:
%p A265735 x1:=evalhf(Pi*n-1/n):y1:=evalhf(Pi*n+1/n):
%p A265735 x:=floor(x1):y:=floor(y1):
%p A265735 for j from x+1 to y do:
%p A265735 printf("%g %g %d %d\n",x1,y1,j,n):
%p A265735 od:
%p A265735 od:
%t A265735 kmax=10^9; Flatten[Table[Range[Ceiling[Pi k-1/k], Floor[Pi k+1/k]], {k, kmax}]] (* or limiting memory usage *)
%t A265735 a = {3,4}; kmax = 10^9; For[k = 1, k <= kmax, k++,
%t A265735  If[(nw = Ceiling[Pi k - 1/k]) == Floor[Pi k + 1/k],
%t A265735   AppendTo[a, nw]]]; a (* _Stefano Spezia_, Apr 26 2023 *)
%Y A265735 Cf. A000796, A265739. Contains A002485 (without the first two terms) as a subsequence.
%K A265735 nonn
%O A265735 1,1
%A A265735 _Michel Lagneau_, Dec 15 2015