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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.

%I A195325 #62 May 01 2024 13:17:04
%S A195325 2,59,71,29,59,149,191,641,149,347,809,461,3371,1487,857,1301,1877,
%T A195325 5849,4721,9239,4271,1619,1481,20507,20981,32117,13337,19379,24977,
%U A195325 48779,20441,25301,5651,37991,17747,43577,176777,145757,191249,84809,150209,11717
%N 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.
%C A195325 Such a prime always exists.
%C A195325 The sequence is unbounded.
%C A195325 Conjecture. For n >= 2, a(n) is a lesser of twin primes (A001359). This implies the twin prime conjecture. - _Vladimir Shevelev_, Sep 15 2011
%C A195325 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
%C A195325 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.
%H A195325 Charles R Greathouse IV, <a href="/A195325/b195325.txt">Table of n, a(n) for n = 1..169</a>, (first 100 terms from Alois P. Heinz)
%H A195325 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%p A195325 a:= proc(n) local p, q;
%p A195325       p:= 2; q:= nextprime(p);
%p A195325       while nextprime(n*p) < (n*q) do
%p A195325         p, q:= q, nextprime(q)
%p A195325       od; p
%p A195325     end:
%p A195325 seq (a(n), n=1..25); # _Alois P. Heinz_, Sep 15 2011
%t A195325 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 *)
%Y A195325 Cf. A080192, A195270, A195271, A164368, A194658, A164294, A110835, A195465.
%K A195325 nonn
%O A195325 1,1
%A A195325 _Vladimir Shevelev_, Sep 15 2011