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A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

%I A031924 #38 Aug 02 2024 12:08:22
%S A031924 23,31,47,53,61,73,83,131,151,157,167,173,233,251,257,263,271,331,353,
%T A031924 367,373,383,433,443,503,541,557,563,571,587,593,601,607,647,653,677,
%U A031924 727,733,751,941,947,971,977,991,1013,1033,1063,1097,1103,1117,1123,1181
%N A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.
%C A031924 Original name: Lower prime of a difference of 6 between consecutive primes.
%C A031924 Conjecture: The sequence is infinite and for every n >= 7746, a(n+1) < a(n)^(1+1/n). Namely for n >= 7746, a(n)^(1/n) is a strictly decreasing function of n (See comment lines of the sequence A248855). - _Jahangeer Kholdi_ and _Farideh Firoozbakht_, Nov 29 2014
%H A031924 Charles R Greathouse IV, <a href="/A031924/b031924.txt">Table of n, a(n) for n = 1..10000</a>
%H A031924 OEIS wiki, <a href="/wiki/Consecutive_primes_in_arithmetic_progression#CPAP_with_given_gap">Consecutive primes in arithmetic progression: CPAP with given gap</a>, updated Jan. 2020
%H A031924 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%e A031924 23 is a term as the next prime 29 = 23 + 6.
%p A031924 A031924 := proc(n)
%p A031924     option remember;
%p A031924     if n = 1 then
%p A031924         return 23;
%p A031924     else
%p A031924         p := nextprime(procname(n-1)) ;
%p A031924         q := nextprime(p) ;
%p A031924         while q-p <> 6 do
%p A031924             p := q ;
%p A031924             q := nextprime(p) ;
%p A031924         end do:
%p A031924         return p;
%p A031924     end if;
%p A031924 end proc: # _R. J. Mathar_, Jan 23 2013
%t A031924 Transpose[Select[Partition[Prime[Range[200]], 2, 1], Last[#] - First[#] == 6 &]][[1]] (* _Bruno Berselli_, Apr 09 2013 *)
%o A031924 (PARI) is(n)=isprime(n)&&nextprime(n+1)-n==6 \\ _Charles R Greathouse IV_, Mar 21 2013
%o A031924 (PARI) apply( A031924(n,p=2,show=0,g=6)={forprime(q=p+1,, p+g!=(p=q) || (show&&print1(p-g",")) || n-- || return(p-g))}, [1..99]) \\ Use nxt(p)=A031924(1,p) to get the term following p, use show=1 to print all a(1..n), g to select a different gap. - _M. F. Hasler_, Jan 02 2020
%o A031924 (Magma) [p: p in PrimesUpTo(1200) | NextPrime(p)-p eq 6]; // _Bruno Berselli_, Apr 09 2013
%o A031924 (GAP) P:=Filtered([1..1200],IsPrime);;
%o A031924 List(Filtered([1..Length(P)-1],i->P[i+1]-P[i]=6),k->P[k]); # _Muniru A Asiru_, Jan 30 2019
%Y A031924 Cf. A001359, A023201, A031925; A031924 and A007529 together give A023201.
%K A031924 nonn
%O A031924 1,1
%A A031924 _Jeff Burch_
%E A031924 New name from _M. F. Hasler_, Jan 02 2020