A282594 - OEIS

%I #12 Feb 20 2017 23:24:12

%S 307,353,409,461,499,509,593,647,673,743,811,863,929,1051,1123,1163,

%T 1201,1217,1279,1453,1553,1657,1697,1783,1823,1889,1907,1931,1973,

%U 2029,2089,2131,2141,2203,2243,2267,2297,2311,2411,2417,2531,2579,2593,2609,2617

%N Primes p > 5 such that odd part of (p^2-q^2)/3 is composite for every prime q, 3 < q < p.

%C If prime(n) is in the sequence, then necessarily A282445(n) = 0. On the other hand, if A282445(n) = 0, then prime(n) is in the sequence if and only if all numbers {odd part of (prime(n)^2-q^2)/3, q is prime, 3 < q < prime(n)} are more than 1.

%H Charles R Greathouse IV, <a href="/A282594/b282594.txt">Table of n, a(n) for n = 1..10000</a>

%e The smallest n for which A282445(n)=0 is 44. Prime(44)=193. For q=5,7,..., 181, odd part of (p^2-q^2)/3 is 4653,775,...,187 respectively which are all composite numbers. But for q=191, we have 1. Therefore, 193 is not in the sequence.

%o (PARI) is(n)=if(!isprime(n), return(0)); my(p2=n^2,t); forprime(q=5,n-2, t=(p2-q^2)/3; t>>=valuation(t,2); if(isprime(t) || t==1, return(0))); n > 5 \\ _Charles R Greathouse IV_, Feb 20 2017

%Y Cf. A000040, A000265, A282445.

%K nonn

%O 1,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Feb 19 2017