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%I #10 Jan 18 2016 10:26:43
%S 1,2,3,4,7,14,32,39,71,142,465,536,1001,2002,3003,8544,9545,18089,
%T 36178,54267,190435,208524,398959,797918,1196877,4996032,5394991,
%U 10391023,20782046,31173069,41564092,51955115
%N 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.
%C Conjecture: the sequence is infinite.
%C See the reference for a similar problem with Fibonacci numbers.
%C 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).
%C For k > 1, the interval [e*k - 1/k, e*k + 1/k] contains exactly one integer.
%C We observe two properties:
%C (1) a(n) = m*a(n-m+1) for some n, m=2,3,4 and 5
%C Examples:
%C 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 m = 3 => a(16)=3*a(14), a(21)=3*a(19), a(26)=3*a(24), a(31)=3*a(29), ...
%C m = 4 => a(4)=4*a(1), a(32)=4*a(29), ...
%C m = 5 => a(33)=5*a(29), ...
%C (ii) a(n+2) = a(n) + a(n+1) for n = 1, 3, 7, 11, 13, 16, 18, 21, 23, 26, 28, ...
%H Takao Komatsu, <a href="http://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.
%e 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...];
%e a(2) = 2 because the number 5 belongs to the interval [2*e-1/2, 2*e+1/2] = [4.936564..., 5.936564...];
%e a(3) = 3 because the number 8 belongs to the interval [3*e-1/3, 3*e+1/3] = [7.821512..., 8.488179...].
%p *** the program gives the interval [a,b],the integer(s) between [a,b] and k ***
%p nn:=10^9:
%p e:=exp(1):
%p for n from 1 to nn do:
%p x1:=evalhf(e*n-1/n):y1:=evalhf(e*n+1/n):
%p x:=floor(x1):y:=floor(y1):
%p for j from x+1 to y do:
%p printf("%g %g %g %g\n",x1,y1,j,n):
%p od:
%p od:
%Y Cf. A001113, A265741.
%K nonn,more
%O 1,2
%A _Michel Lagneau_, Dec 15 2015
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