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

2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, 784, 786, 788, 904, 906, 908, 1114, 1116, 1118, 1144, 1146, 1148, 1264, 1266, 1268, 1354, 1356, 1358, 3244, 3246, 3248, 4204, 4206, 4208

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• nice

Comments

Conjectured to be complete (although if this were proved it would prove the "twin primes conjecture"!).

No other n < 10^9. - T. D. Noe, Apr 10 2007

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

Examples

			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.
		

References

• Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 132.

Links

• Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
• James Grime and Brady Haran, Goldbach Conjecture (but with TWIN PRIMES), Numberphile video (2024)
• Eric Weisstein's World of Mathematics, Twin Primes
• Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A051345, A179825, A129363 (number of partitions of 2n into the sum of two twin primes).

A064409 is a different sequence with a superficially similar definition.

Programs

Haskell

import qualified Data.Set as Set (map, null)
import Data.Set (empty, insert, intersection)
a007534 n = a007534_list !! (n-1)
a007534_list = f [2,4..] empty 1 a001097_list where
   f xs'@(x:xs) s m ps'@(p:ps)
     | x > m = f xs' (insert p s) p ps
     | Set.null (s `intersection` Set.map (x -) s) = x : f xs s m ps'
     | otherwise = f xs s m ps'
-- Reinhard Zumkeller, Nov 27 2011

Mathematica

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} ] ] ] ]
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 *)