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A118955
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Numbers of the form 2^k + prime.
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19
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3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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The lower density is at least 0.09368 (Pintz) and upper density is at most 0.49095 (Habsieger & Roblot). The density, if it exists, is called Romanov's constant. Romani conjectures that it is around 0.434. - Charles R Greathouse IV, Mar 12 2008
Elsholtz & Schlage-Puchta improve the bound on lower density to 0.107648. Unpublished work by Jie Wu improves this to 0.110114, see Remark 1 in Elsholtz & Schlage-Puchta. - Charles R Greathouse IV, Aug 06 2021
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LINKS
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Christian Elsholtz and Jan-Christoph Schlage-Puchta, On Romanov's constant, Mathematische Zeitschrift, Vol. 288 (2018), pp. 713-724.
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MATHEMATICA
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PROG
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(PARI) list(lim)=my(v=List(), t=1); while(t<lim, forprime(p=2, lim-t, listput(v, p+t)); t<<=1); Set(v) \\ Charles R Greathouse IV, Aug 06 2021
(Haskell)
a118955 n = a118955_list !! (n-1)
a118955_list = filter f [1..] where
f x = any (== 1) $ map (a010051 . (x -)) $ takeWhile (< x) a000079_list
(Python)
from itertools import count, islice
from sympy import isprime
def A118955_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n: any(isprime(n-(1<<i)) for i in range(n.bit_length()-1, -1, -1)), count(max(startvalue, 1)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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