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A277688
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Odd numbers k such that there is no prime p < k/2 with k - 2*p and k + 2*p both prime.
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3
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1, 3, 5, 19, 29, 31, 43, 49, 55, 59, 61, 71, 79, 83, 89, 91, 101, 109, 113, 115, 119, 125, 127, 131, 139, 149, 151, 155, 161, 163, 167, 169, 175, 179, 191, 193, 197, 199, 203, 209, 211, 215, 223, 227, 229, 239, 241, 247, 251, 253, 259, 265, 269, 271, 281, 283
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OFFSET
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1,2
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COMMENTS
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Or, odd integers k such that k + 2*p is composite for all primes p, q with 2*p + q = k. By the Lemoine-Levy conjecture, for every odd k>5, there are primes p and q such that k=2*p+q. Numbers 1,3,5 formally satisfy the condition.
The sequence is an analog of A284919 for odd numbers.
Conjecture: k=59 and k=151 are the only terms k>5 satisfying the additional condition that k + 2*q is composite for every prime p,q such that 2*p+q=k.
More than half of all odd numbers are in this sequence: for k < 2000, the percentage is below 50%, but for k < 1e4, 2e4 and 4e4 the percentage is > 55%, 56% and 58%, respectively. - M. F. Hasler, Apr 11 2017
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LINKS
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MATHEMATICA
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Select[Range[1, 283, 2], Total@ Boole@ Map[Function[p, Times @@ Boole@ Map[PrimeQ, {# - 2 p, # + 2 p}] == 1], Prime@ Range@ PrimePi[#/2]] == 0 &] (* Michael De Vlieger, Apr 22 2017 *)
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PROG
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(PARI) is(k)=bittest(k, 0)&&!forprime(p=2, k\2, (isprime(k-2*p)&&isprime(k+2*p))&&return) \\ M. F. Hasler, Apr 11 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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