A038773 a(n) is the smallest prime of the form Q + c, where Q is the n-th primorial and c is a composite >= prime(n+1)^2.

11, 31, 79, 331, 2531, 30319, 511039, 9700357, 223093769, 6469694377, 200560491721, 7420738136831, 304250263529059, 13082761331672803, 614889782588494961, 32589158477190048817, 1922760350154212643889, 117288381359406970988027, 7858321551080267055884131, 557940830126698960967422909

Offset: 1

Author

Labos Elemer, May 04 2000

Keywords

• nonn

Comments

Between 2310 and 2531 there are 26 primes (2311, ..., 2521), all of which are of the form (primorial + prime). (2311 = 2 + 2309 (prime) = 2*3*5 + 2281 (prime); each of the other 25 primes is of the form 2*3*5*7*11 + prime.)

Observe that a(2) = 31 = 2*3 + 5^2 = 2*3*5 + 1, so it has two "primorial forms".

Examples

			At n=5, the 5th primorial is A002110(5)=2310 and 2310 + 13*17 = 2310 + 221 = 2531 is the prime that meets the criteria of the definition.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A002110, A054757, A054758, A005235.

Programs

Mathematica

Array[Block[{Q = Product[Prime@ i, {i, #}], c = Prime[# + 1]^2}, While[Nand[PrimeQ[Q + c], CompositeQ@ c], c++]; Q + c] &, 17] (* Michael De Vlieger, May 22 2018 *)

PARI

a(n) = {my(pr = prod(k=1, n, prime(k)), c = prime(n+1)^2); while (isprime(c) || !isprime(pr + c), c++); pr + c;} \\ Michel Marcus, May 26 2018

Extensions

Edited by Jon E. Schoenfield, May 22 2018

More terms from Michael De Vlieger, May 22 2018