A282594 Primes p > 5 such that odd part of (p^2-q^2)/3 is composite for every prime q, 3 < q < p.

307, 353, 409, 461, 499, 509, 593, 647, 673, 743, 811, 863, 929, 1051, 1123, 1163, 1201, 1217, 1279, 1453, 1553, 1657, 1697, 1783, 1823, 1889, 1907, 1931, 1973, 2029, 2089, 2131, 2141, 2203, 2243, 2267, 2297, 2311, 2411, 2417, 2531, 2579, 2593, 2609, 2617

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Feb 19 2017

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000265, A282445.

Programs

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