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Odd numbers k such that there are no consecutive prime numbers p, q such that 2*(p - k) = q + k.
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%I #22 Dec 07 2019 01:49:13

%S 7,15,17,29,37,39,47,51,55,61,67,69,71,81,83,85,87,95,97,99,105,107,

%T 111,113,119,121,123,129,135,141,149,155,159,163,167,169,171,175,177,

%U 181,183,185,187,191,193,195,197,201,209,211,215,217,221,229,235,239,241,243,247,249

%N Odd numbers k such that there are no consecutive prime numbers p, q such that 2*(p - k) = q + k.

%C The complementary sequence that include the numbers k which do satisfy 2*(p - k) = q + k is {1, 3, 5, 9, 11, 13, 19, 21, 23, 25, 27, 31, 33, 35, 41, 43, 45, 49, 53, 57, 59, ...}. The number 7 does not satisfy the formula, so is the first term of sequence.

%C The terms of the sequence are the multiples of 3 not included in A062234 and divided by 3.

%e 7 is a term because 2*(p - 7) <> q + 7 for every p, q consecutive prime numbers. See comments.

%o (PARI) v=vector(1000); forstep(k=1,299,2,forprime(n=2,1000,p=nextprime(n+1); if(2*(n-k)==p+k,v[k]=1;break))); forstep(k=1,250,2,if(v[k]==0,print1(k", ")))

%Y Cf. A000040, A062234.

%K nonn

%O 1,1

%A _Dimitris Valianatos_, Nov 14 2019