

A007534


Even numbers that are not the sum of a pair of twin primes.
(Formerly M1306)


16



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
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OFFSET

1,1


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


REFERENCES

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 19992000.
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

Table of n, a(n) for n=1..35.
Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 19992000.
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


EXAMPLE

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.


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 *)


PROG

(Haskell)
import qualified Data.Set as Set (map, null)
import Data.Set (empty, insert, intersection)
a007534 n = a007534_list !! (n1)
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


CROSSREFS

Cf. A051345.
Cf. A129363 (number of partitions of 2n into the sum of two twin primes).
Cf. A179825.
Sequence in context: A018410 A270484 A156496 * A009379 A092918 A018428
Adjacent sequences: A007531 A007532 A007533 * A007535 A007536 A007537


KEYWORD

nonn,nice


AUTHOR

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


STATUS

approved



