A277688 - OEIS

%I #34 Apr 22 2017 10:27:31

%S 1,3,5,19,29,31,43,49,55,59,61,71,79,83,89,91,101,109,113,115,119,125,

%T 127,131,139,149,151,155,161,163,167,169,175,179,191,193,197,199,203,

%U 209,211,215,223,227,229,239,241,247,251,253,259,265,269,271,281,283

%N Odd numbers k such that there is no prime p < k/2 with k - 2*p and k + 2*p both prime.

%C 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.

%C The sequence is an analog of A284919 for odd numbers.

%C 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.

%C This conjecture arose from the calculations up to 500001 by _Peter J. C. Moses_ and confirmed by _M. F. Hasler_.

%C 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

%t 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 *)

%o (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

%Y Cf. A046927, A284919.

%K nonn

%O 1,2

%A _Vladimir Shevelev_, Apr 11 2017

%E More terms from _Peter J. C. Moses_, Apr 11 2017