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A007534 Positive even numbers that are not the sum of a pair of twin primes.

%I A007534 M1306 #58 Feb 16 2025 08:32:31
%S A007534 2,4,94,96,98,400,402,404,514,516,518,784,786,788,904,906,908,1114,
%T A007534 1116,1118,1144,1146,1148,1264,1266,1268,1354,1356,1358,3244,3246,
%U A007534 3248,4204,4206,4208
%N A007534 Positive even numbers that are not the sum of a pair of twin primes.
%C A007534 Conjectured to be complete (although if this were proved it would prove the "twin primes conjecture"!).
%C A007534 No other n < 10^9. - _T. D. Noe_, Apr 10 2007
%C A007534 Of these 35, the only 5 which are two times a prime (or in A001747) are 4 = 2 * 2, 94 = 2 * 47, 514 = 2 * 257, 1114 = 2 * 557, 1354 = 2 * 677. - _Jonathan Vos Post_, Mar 06 2010
%D A007534 Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
%D A007534 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007534 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 132.
%H A007534 Harvey Dubner, <a href="/A007534/a007534.pdf">Twin Prime Conjectures</a>, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
%H A007534 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=Gojd8mTl3Do">Goldbach Conjecture (but with TWIN PRIMES)</a>, Numberphile video (2024)
%H A007534 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H A007534 Dan Zwillinger, <a href="http://www.ams.org/journals/mcom/1979-33-147/S0025-5718-1979-0528060-5/S0025-5718-1979-0528060-5.pdf">A Goldbach Conjecture Using Twin Primes</a>, Math. Comp. 33, No.147 (1979), p.1071.
%H A007534 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A007534 The twin primes < 100 are 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73. 94 is in the sequence because no combination of any two numbers from the set just enumerated can be summed to make 94.
%t A007534 p = Select[ Range[ 4250 ], PrimeQ[ # ] && PrimeQ[ # + 2 ] & ]; q = Union[ Join[ p, p + 2 ] ]; Complement[ Table[ n, {n, 2, 4250, 2} ], Union[ Flatten[ Table[ q[ [ i ] ] + q[ [ j ] ], {i, 1, 223}, {j, 1, 223} ] ] ] ]
%t A007534 Complement[Range[2,4220,2],Union[Total/@Tuples[Union[Flatten[ Select[ Partition[ Prime[ Range[500]],2,1],#[[2]]-#[[1]]==2&]]],2]]] (* _Harvey P. Dale_, Oct 09 2013 *)
%o A007534 (Haskell)
%o A007534 import qualified Data.Set as Set (map, null)
%o A007534 import Data.Set (empty, insert, intersection)
%o A007534 a007534 n = a007534_list !! (n-1)
%o A007534 a007534_list = f [2,4..] empty 1 a001097_list where
%o A007534    f xs'@(x:xs) s m ps'@(p:ps)
%o A007534      | x > m = f xs' (insert p s) p ps
%o A007534      | Set.null (s `intersection` Set.map (x -) s) = x : f xs s m ps'
%o A007534      | otherwise = f xs s m ps'
%o A007534 -- _Reinhard Zumkeller_, Nov 27 2011
%Y A007534 Cf. A051345, A179825, A129363 (number of partitions of 2n into the sum of two twin primes).
%Y A007534 A064409 is a different sequence with a superficially similar definition.
%K A007534 nonn,nice
%O A007534 1,1
%A A007534 _N. J. A. Sloane_, _Robert G. Wilson v_