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A007534 Positive even numbers that are not the sum of a pair of twin primes.
(Formerly M1306)
18
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
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), 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.
Eric Weisstein's World of Mathematics, Twin Primes
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, A179825, A129363 (number of partitions of 2n into the sum of two twin primes).
A064409 is a different sequence with a superficially similar definition.
Sequence in context: A335571 A335291 A156496 * A299784 A009379 A092918
KEYWORD
nonn,nice
AUTHOR
STATUS
approved

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Last modified March 18 21:02 EDT 2024. Contains 370951 sequences. (Running on oeis4.)