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).

3, 23, 139, 293, 1129, 2477, 8467, 30593, 81463, 85933, 190409, 404597, 535399, 840353, 1100977, 2127163, 4640599, 6613631, 6958667, 10343761, 24120233, 49269581, 83751121, 101649649, 166726367, 273469741, 310845683, 568951459

Offset: 1

Author

Vladimir Shevelev, Jan 06 2011

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n.

Crossrefs

Cf. A168253, A179210, A179234, A179240, A179256, A001223

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local k, p, q, r, pn;
      pn:= ithprime(n);
      for k from `if`(n=1, 1, pi(a(n-1))) do
        p:= ithprime(k);
        q:= ithprime(k+1);
        r:= ithprime(k+2);
        if denom((q-p)/(r-q)) = pn then break fi
      od; q
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Jan 06 2011

Mathematica

a[n_] := a[n] = Module[{k, p, q, r, pn},
     pn = Prime[n];
     For[k = If[n == 1, 1, PrimePi[a[n - 1]]], True, k++,
     p = Prime[k];
     q = Prime[k + 1];
     r = Prime[k + 2];
     If [Denominator[(q - p)/(r - q)] == pn, Break[]]]; q];
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Jan 06 2011