A359702 Odd primes p that are not congruent to 2*k modulo prime(k+1) for any positive integer k.

3, 7, 31, 37, 43, 61, 67, 73, 157, 211, 271, 277, 331, 367, 421, 457, 571, 691, 823, 883, 997, 1093, 1201, 1237, 1303, 1657, 1783, 2053, 2287, 2347, 2371, 2377, 2557, 2803, 2971, 3001, 3061, 3067, 3307, 3313, 3391, 3967, 4021, 4231, 4273, 4357, 4447, 4561, 4603

Offset: 1

Author

Andrea La Rosa, Jan 11 2023

Keywords

• nonn

Comments

This sequence arises from a more general study. First, consider a function f : P -> N (where P is the set of the odd prime numbers) such that 0 <= f(p) < p. Then, remove from the set P each prime number q such that q = f(p) (mod p) for some p.

For example, if f(p) = 0 for each p, then the final set is the empty set.

If f(p) = 1 for each p, then the final set seems to be the set of Fermat primes (empirical observation).

If f(p) = p-1, then the final set seems to be the set of Mersenne primes (empirical observation).

For the particular choice f(p) = 2k (where p is the k-th odd prime) this sequence is obtained.

Examples

			Terms in this sequence are those odd primes that are neither congruent to 2 (mod 3), nor congruent to 4 (mod 5), nor congruent to 6 (mod 7), nor congruent to 8 (mod 11), etc.
7 is a term because 7 == 1 (mod 3) and 7 == 2 (mod 5).
11 is not a term because 11 == 2 (mod 3).
13 is not a term because 13 == 6 (mod 7).
17 is not a term because 17 == 2 (mod 3).
19 is not a term because 19 == 8 (mod 11).
		

Links

• Mathematics StackExchange, Can the set of odd primes be finitely sieved by arbitrary congruences of primes?

Crossrefs

Cf. A019434, A000668.

Programs

PARI

isok(p) = {if(!isprime(p)||p==2, 0, my(k=0); forprime(q=3, p-1, k+=2; if(p%q==k, return(0))); 1) } \\ Andrew Howroyd, Jan 11 2023

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 11 2023