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A179328 a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator prime(n) (or 0, if such a prime does not exist).

%I A179328 #18 Mar 18 2022 10:27:10
%S A179328 3,23,139,293,1129,2477,8467,30593,81463,85933,190409,404597,535399,
%T A179328 840353,1100977,2127163,4640599,6613631,6958667,10343761,24120233,
%U A179328 49269581,83751121,101649649,166726367,273469741,310845683,568951459
%N A179328 a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator prime(n) (or 0, if such a prime does not exist).
%C A179328 Conjecture: a(n) > 0 for all n.
%p A179328 with(numtheory):
%p A179328 a:= proc(n) option remember; local k, p, q, r, pn;
%p A179328       pn:= ithprime(n);
%p A179328       for k from `if`(n=1, 1, pi(a(n-1))) do
%p A179328         p:= ithprime(k);
%p A179328         q:= ithprime(k+1);
%p A179328         r:= ithprime(k+2);
%p A179328         if denom((q-p)/(r-q)) = pn then break fi
%p A179328       od; q
%p A179328     end:
%p A179328 seq(a(n), n=1..10);  # _Alois P. Heinz_, Jan 06 2011
%t A179328 a[n_] := a[n] = Module[{k, p, q, r, pn},
%t A179328      pn = Prime[n];
%t A179328      For[k = If[n == 1, 1, PrimePi[a[n - 1]]], True, k++,
%t A179328      p = Prime[k];
%t A179328      q = Prime[k + 1];
%t A179328      r = Prime[k + 2];
%t A179328      If [Denominator[(q - p)/(r - q)] == pn, Break[]]]; q];
%t A179328 Table[a[n], {n, 1, 10}] (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%Y A179328 Cf. A168253, A179210, A179234, A179240, A179256, A001223
%K A179328 nonn
%O A179328 1,1
%A A179328 _Vladimir Shevelev_, Jan 06 2011
%E A179328 More terms from _Alois P. Heinz_, Jan 06 2011