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%I #18 Mar 18 2022 10:27:10
%S 3,23,139,293,1129,2477,8467,30593,81463,85933,190409,404597,535399,
%T 840353,1100977,2127163,4640599,6613631,6958667,10343761,24120233,
%U 49269581,83751121,101649649,166726367,273469741,310845683,568951459
%N 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 Conjecture: a(n) > 0 for all n.
%p with(numtheory):
%p a:= proc(n) option remember; local k, p, q, r, pn;
%p pn:= ithprime(n);
%p for k from `if`(n=1, 1, pi(a(n-1))) do
%p p:= ithprime(k);
%p q:= ithprime(k+1);
%p r:= ithprime(k+2);
%p if denom((q-p)/(r-q)) = pn then break fi
%p od; q
%p end:
%p seq(a(n), n=1..10); # _Alois P. Heinz_, Jan 06 2011
%t a[n_] := a[n] = Module[{k, p, q, r, pn},
%t pn = Prime[n];
%t For[k = If[n == 1, 1, PrimePi[a[n - 1]]], True, k++,
%t p = Prime[k];
%t q = Prime[k + 1];
%t r = Prime[k + 2];
%t If [Denominator[(q - p)/(r - q)] == pn, Break[]]]; q];
%t Table[a[n], {n, 1, 10}] (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%Y Cf. A168253, A179210, A179234, A179240, A179256, A001223
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Jan 06 2011
%E More terms from _Alois P. Heinz_, Jan 06 2011
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