cp's OEIS Frontend

A187072 Prime numbers chosen such that the even numbers that are the sum of two consecutive terms occur only once and occur as early as possible.

%I A187072 #35 Mar 13 2023 10:21:33
%S A187072 3,3,5,5,7,7,11,5,17,3,23,5,19,11,23,13,19,19,23,17,29,19,31,13,41,11,
%T A187072 47,13,43,19,47,17,53,19,59,17,67,7,61,19,67,23,59,29,67,31,61,41,53,
%U A187072 47,59,53,61,43,67,41,79,37,89,29,101,23,109,13,127,7,131,5,137,7,139,11,137,17,139,13,149,11,157,7,151,19,109,67,107,59,113,67
%N A187072 Prime numbers chosen such that the even numbers that are the sum of two consecutive terms occur only once and occur as early as possible.
%C A187072 The even numbers a(n) + a(n+1) are in sequence A187085.
%C A187072 The terms for even n grow rapidly; for odd n they grow slowly. It appears that primes occur at a consistent frequency: in the first 1000000 terms, primes 3 to 23 occur about 4.7%, 4.9%, 3.4%, 2.9%, 2.6%, 2.0%, 1.8%, and 1.4% of the time. - _T. D. Noe_, Mar 04 2011
%H A187072 T. D. Noe, <a href="/A187072/b187072.txt">Table of n, a(n) for n = 1..5000</a>
%H A187072 T. D. Noe, <a href="/A187072/a187072.gif">Plot of frequency (%) of primes in the first 10^9 terms</a>
%H A187072 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A187072 Primes: 3 3 5  5  7  7  11  5  17  3  23  5  19  11  23  13  19  19  23
%e A187072 Evens:   6 8 10 12 14 18  16 22  20 26  28 24  30  34  36  32  38  42
%t A187072 lastE=10; eList=Range[6,lastE,2]; evens[k_] := If[k<=Length[eList], eList[[k]], lastE+=2; AppendTo[eList,lastE]; lastE]; Join[{lastP=3}, Table[k=1; While[p=evens[k]-lastP; p<0 || !PrimeQ[p], k++]; eList=Delete[eList,k]; lastP=p, {999}]] (* _T. D. Noe_, Mar 04 2011 *)
%t A187072 s={3,3}; ev={6}; a=3; Do[k=2; While[!FreeQ[ev,(b=a+(p=Prime[k]))],k++]; a=p; AppendTo[ev,b]; AppendTo[s,a], {3000}]; s (* Zak Seidov, Mar 03 2011 *)
%o A187072 (Haskell)
%o A187072 import Data.Set (Set, empty, member, insert)
%o A187072 a187072 n = a187072_list !! (n-1)
%o A187072 a187072_list = goldbach 0 a065091_list empty where
%o A187072   goldbach :: Integer -> [Integer] -> Set Integer -> [Integer]
%o A187072   goldbach q (p:ps) gbEven
%o A187072       | qp `member` gbEven = goldbach q ps gbEven
%o A187072       | otherwise          = p : goldbach p a065091_list (insert qp gbEven)
%o A187072       where qp = q + p
%o A187072 -- performance bug fixed: _Reinhard Zumkeller_, Mar 06 2011
%Y A187072 Cf. A065091, A118371, A187085, A187098.
%K A187072 nonn,look
%O A187072 1,1
%A A187072 _Reinhard Zumkeller_, Mar 03 2011