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A168253 a(n) is the smallest prime q such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator n (or 0, if such a prime does not exist).

%I A168253 #20 Dec 11 2016 15:52:37
%S A168253 5,3,23,89,139,199,113,1933,523,3089,1129,1669,2477,2971,4297,5591,
%T A168253 1327,28351,30593,19333,16141,36389,81463,28229,31907,19609,35617,
%U A168253 82073,44293,102701,34061,288583,221327,134513,173359,360091
%N A168253 a(n) is the smallest prime q such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator n (or 0, if such a prime does not exist).
%C A168253 Conjecture: a(n)>0 for all n.
%H A168253 Robert G. Wilson v, <a href="/A168253/b168253.txt">Table of n, a(n) for n = 1..127</a>
%t A168253 f[n_] := Block[{p = 2, q = 3, r = 5}, While[ Numerator[(r - q)/(q - p)] != n, p = q; q = r; r = NextPrime@ r]; q]; Array[f, 36]
%t A168253 p = 2; q = 3; r = 5; t[_] = 0; While[q < 100000000, If[ t[ Denominator[(q - p)/(r - q)]] == 0, t[ Denominator[(q - p)/(r - q)]] = q]; p = q; q = r; r = NextPrime@ r]; t@# & /@ Range@100 (* _Robert G. Wilson v_, Dec 11 2016 *)
%Y A168253 Cf. A179210, A179234, A001223
%K A168253 nonn
%O A168253 1,1
%A A168253 _Vladimir Shevelev_, Jan 05 2011