A195325 Least n-gap prime: a(n) = least prime p for which there is no prime between n*p and n*q, where q is the next prime after p.

2, 59, 71, 29, 59, 149, 191, 641, 149, 347, 809, 461, 3371, 1487, 857, 1301, 1877, 5849, 4721, 9239, 4271, 1619, 1481, 20507, 20981, 32117, 13337, 19379, 24977, 48779, 20441, 25301, 5651, 37991, 17747, 43577, 176777, 145757, 191249, 84809, 150209, 11717

Offset: 1

Author

Vladimir Shevelev, Sep 15 2011

Keywords

• nonn

Comments

Such a prime always exists.

The sequence is unbounded.

Conjecture. For n >= 2, a(n) is a lesser of twin primes (A001359). This implies the twin prime conjecture. - Vladimir Shevelev, Sep 15 2011

If a member of this sequence is not the lesser of a twin prime pair, it is greater than 10^10. - Charles R Greathouse IV, Sep 15 2011

A dual sequence: b(n)= least prime p for which there is no prime between n*q and n*p, where q is the previous prime before p. Evidently, b(n) is the next prime after a(n): 3,61,73,31,..., and for n>=2, by the same conjecture, b(n) is a greater of twin primes.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..169, (first 100 terms from Alois P. Heinz)
• Index entries for primes, gaps between

Crossrefs

Cf. A080192, A195270, A195271, A164368, A194658, A164294, A110835, A195465.

Programs

Maple

a:= proc(n) local p, q;
      p:= 2; q:= nextprime(p);
      while nextprime(n*p) < (n*q) do
        p, q:= q, nextprime(q)
      od; p
    end:
seq (a(n), n=1..25); # Alois P. Heinz, Sep 15 2011

Mathematica

pQ[p_, r_] := Block[{q = NextPrime[p]},NextPrime[r*p]> r*q]; f[n_] := Block[{p = 2}, While[ !pQ[p, n], p = NextPrime[p]]; p]; f[1] = 2; Array[f, 42] (* Robert G. Wilson v, Sep 18 2011 *) (* Revised by Zak Seidov, Sep 19 2011 *)