A031930 - OEIS

%I #34 Apr 30 2024 05:49:25

%S 199,211,467,509,619,661,797,997,1201,1237,1307,1459,1499,1511,1531,

%T 1709,1789,1811,1889,2069,2099,2297,2399,2447,2579,2621,2777,2927,

%U 3049,3067,3169,3191,3259,3331,3347,3499,3559,3659,3931,3989

%N Lower prime of a difference of 12 between consecutive primes.

%C Some of the terms of this sequence are primes that are separated from both their predecessor and successor primes by 12, e.g., 211, 1511, 4409, 4691, 7841, 9871, 11299, 11411, 11731.  See A053072. - _Harvey P. Dale_, Apr 07 2013

%C Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely 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

%C Aside from 2 and 3, all primes are congruent to 1, 5, 7, 11 mod 12. Thus the least significant duodecimal digit of any term in this sequence is 1, 5, 7 or B. - _Alonso del Arte_, Aug 19 2017

%H Charles R Greathouse IV, <a href="/A031930/b031930.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%F a(n) = prime(A320704(n)). - _R. J. Mathar_, Apr 30 2024

%e 199 is a term as the next prime is 199 + 12 = 211.

%e 211 is also a term since the next prime is 211 + 12 = 223.

%e But 223 is not a term since the next prime is 227, and 223 + 12 = 235 = 5 * 47.

%t Transpose[Select[Partition[Prime[Range[600]], 2, 1], Last[#] - First[#] == 12 &]][[1]] (* _Harvey P. Dale_, Apr 07 2013 *)

%o (Magma) [p: p in PrimesUpTo(4000) | NextPrime(p)-p eq 12]; // _Bruno Berselli_, Apr 09 2013

%o (PARI) is(n)=nextprime(n+1)==n+12 && isprime(n) \\ _Charles R Greathouse IV_, Jul 02 2013

%Y Cf. A053072, A248855, A031928.

%K nonn

%O 1,1

%A _Jeff Burch_