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A007534
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Positive even numbers that are not the sum of a pair of twin primes.
(Formerly M1306)
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18
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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
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1,1
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COMMENTS
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Conjectured to be complete (although if this were proved it would prove the "twin primes conjecture"!).
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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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'
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CROSSREFS
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A064409 is a different sequence with a superficially similar definition.
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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