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A195270 3-gap primes: Prime p is a term iff there is no prime between 3*p and 3*q, where q is the next prime after p.

%I A195270 #29 Nov 29 2023 19:11:40
%S A195270 71,107,137,281,347,379,443,461,557,617,641,727,809,827,853,857,991,
%T A195270 1031,1049,1091,1093,1289,1297,1319,1433,1489,1579,1607,1613,1697,
%U A195270 1747,1787,1867,1871,1877,1931,1987,1997,2027,2237,2269,2309,2377,2381,2473,2591
%N A195270 3-gap primes: Prime p is a term iff there is no prime between 3*p and 3*q, where q is the next prime after p.
%C A195270 For a real r>1, a prime p is called an r-gap prime, if there is no prime between r*p and r*q, where q is the next prime after p. In particular, 2-gap primes are in A080192.
%C A195270 In many cases, q=p+2. E.g., among first 1000 terms there are 509 such cases. - _Zak Seidov_, Jun 29 2015
%H A195270 Zak Seidov, <a href="/A195270/b195270.txt">Table of n, a(n) for n = 1..10000</a>
%p A195270 filter:= p -> isprime(p) and nextprime(3*p)>3*nextprime(p):
%p A195270 select(filter, [2,seq(2*i+1,i=1..2000)]); # _Robert Israel_, Jun 29 2015
%t A195270 pQ[p_, r_] := Block[{q = NextPrime@ p}, Union@ PrimeQ@ Range[r*p, r*q] == {False}]; Select[ Prime@ Range@ 380, pQ[#, 3] &] (* _Robert G. Wilson v_, Sep 18 2011 *)
%t A195270 k = 3; p = 71; Reap[Do[While[NextPrime[k*p] < k*(q = NextPrime[p]), p = q]; Sow[p]; p = q, {1000}]][[2, 1]] (* for first 1000 terms. - _Zak Seidov_, Jun 29 2015 *)
%t A195270 Prime/@SequencePosition[PrimePi[3*Prime[Range[400]]],{x_,x_}][[;;,1]] (* _Harvey P. Dale_, Nov 29 2023 *)
%Y A195270 Cf. A080192, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.
%K A195270 nonn
%O A195270 1,1
%A A195270 _Vladimir Shevelev_, Sep 14 2011