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%I #28 Oct 21 2015 17:24:12
%S 4,5,7,9,13,15,19,7,5,31,11,13,43,45,49,55,61,21,23,73,25,81,85,91,99,
%T 103,21,109,111,115,43,7,139,3,151,51,53,55,169,175,181,61,193,65,199,
%U 201,71,225,229,11,235,241,243,253,259,265,271,13,279,283,285,59,309,313,315,29,37,113,349,27
%N Denominator of (prime(prime(n))+2)/(prime(n)+2).
%C We note that a(n) > 1 for all n = 1..10^7 except for n = 394.
%C Conjecture: For each k = 2,3,6,7,8,9 all the rational numbers (prime(p)+k)/(p+k) with p prime are pairwise distinct.
%C We have verified that all the numbers (prime(p)+2)/(p+2) with p prime and p < 10^7 are indeed pairwise distinct. Also, for each k = 3,6,7,8,9 we have verified that all the numbers (prime(p)+k)/(p+k) with p prime and p < 3*10^6 are indeed pairwise distinct.
%H Zhi-Wei Sun, <a href="/A263427/b263427.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;313507b8.1510">Conjectures on the prime-counting function</a>, a message to Number Theory Mailing List, Oct. 19, 2015.
%e a(1) = 4 since (prime(prime(1))+2)/(prime(1)+2) = (prime(2)+2)/(2+2) = (3+2)/(2+2) = 5/4.
%e a(394) = 1 since (prime(prime(394))+2)/(prime(394)+2) = (prime(2707)+2)/(2707+2) = (24379+2)/(2707+2) = 24381/2709 = 9.
%t p[n_]:=p[n]=Prime[n]
%t a[n_]:=a[n]=Denominator[(p[p[n]]+2)/(p[n]+2)]
%t Table[a[n],{n,1,70}]
%o (PARI) a(n) = denominator((prime(prime(n))+2)/(prime(n)+2));
%o vector(100, n, a(n)) \\ _Altug Alkan_, Oct 18 2015
%Y Cf. A000040, A001359, A006450, A006512, A263399.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Oct 17 2015