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

1,1

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)

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 *)

Crossrefs

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

Sequence in context: A181866 A106897 A244269 * A195329 A197185 A232848

Adjacent sequences: A195322 A195323 A195324 * A195326 A195327 A195328

Keyword

nonn

Author

Vladimir Shevelev, Sep 15 2011

Status

approved