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A007534 Even numbers which are not the sum of a pair of twin primes.
(Formerly M1306)
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

Of these 35, the only 5 which are "even primes" (i.e. 2 times a prime, or A001747) are: 4 = 2 * 2, 94 = 2 * 47, 514 = 2 * 257, 1114 = 2 * 557, 1354 = 2 * 677. [From Jonathan Vos Post, Mar 06 2010]

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

Table of n, a(n) for n=1..35.

Eric Weisstein's World of Mathematics, Twin Primes

Index entries for sequences related to Goldbach conjecture

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.

Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.

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 !! (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

CROSSREFS

Cf. A051345.

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

Cf. A179825.

Sequence in context: A009277 A018410 A156496 * A009379 A092918 A018428

Adjacent sequences:  A007531 A007532 A007533 * A007535 A007536 A007537

KEYWORD

nonn,nice,fini

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified October 22 04:03 EDT 2014. Contains 248388 sequences.