A187072 - OEIS

%I #35 Mar 13 2023 10:21:33

%S 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 47,13,43,19,47,17,53,19,59,17,67,7,61,19,67,23,59,29,67,31,61,41,53,

%U 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 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 The even numbers a(n) + a(n+1) are in sequence A187085.

%C 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 T. D. Noe, <a href="/A187072/b187072.txt">Table of n, a(n) for n = 1..5000</a>

%H T. D. Noe, <a href="/A187072/a187072.gif">Plot of frequency (%) of primes in the first 10^9 terms</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%e Primes: 3 3 5  5  7  7  11  5  17  3  23  5  19  11  23  13  19  19  23

%e Evens:   6 8 10 12 14 18  16 22  20 26  28 24  30  34  36  32  38  42

%t 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 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 (Haskell)

%o import Data.Set (Set, empty, member, insert)

%o a187072 n = a187072_list !! (n-1)

%o a187072_list = goldbach 0 a065091_list empty where

%o   goldbach :: Integer -> [Integer] -> Set Integer -> [Integer]

%o   goldbach q (p:ps) gbEven

%o       | qp `member` gbEven = goldbach q ps gbEven

%o       | otherwise          = p : goldbach p a065091_list (insert qp gbEven)

%o       where qp = q + p

%o -- performance bug fixed: _Reinhard Zumkeller_, Mar 06 2011

%Y Cf. A065091, A118371, A187085, A187098.

%K nonn,look

%O 1,1

%A _Reinhard Zumkeller_, Mar 03 2011