A038771 a(n) is the smallest composite number c such that A002110(n) + c is prime.

4, 9, 25, 49, 121, 221, 289, 529, 667, 899, 1147, 1591, 2021, 1849, 2773, 3551, 4087, 4819, 4757, 5041, 7519, 7663, 8549, 9991, 10379, 13231, 11227, 14659, 11881, 21877, 25283, 18209, 22331, 20989, 22499, 25591, 27221, 29503, 31313, 34547

Offset: 0

Author

Labos Elemer, May 04 2000

Keywords

• nonn

Comments

The lower "envelope" of the sequence is prime(n+1)^2. See also Fortune-conjecture (A005235).

For some n, c=prime(n+1)^2; for others, it is larger, even not necessarily divisible by prime(n+1). E.g., at n=11, prime(11)=31 and a(11) = 1591 = 37*43 = prime(12)*prime(14), while for n=59, a(59) = 97969 = 313^2 = prime(65)^2, etc. Adding these to the suitable primorial numbers, primes are obtained.

Conjecture: lim inf_{n->oo} a(n)/prime(n+1)^2 = 1 < lim sup_{n->oo} a(n)/prime(n+1)^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015

Conjecture: all the terms in this sequence have exactly two prime factors. This conjecture is true for the first 133 terms. - Dmitry Kamenetsky, Jan 06 2019

Links

• Dmitry Kamenetsky, Table of n, a(n) for n = 0..133

Crossrefs

Cf. A002110, A054757, A054758, A005235.

Programs

PARI

a(n) = {my(q = prod(i=1, n, prime(i))); forcomposite(c = 1,, if (isprime(q+c), return(c);););} \\ Michel Marcus, May 24 2015

Extensions

Name edited by Tom Edgar, Jun 08 2015

a(0) prepended by Dmitry Kamenetsky, Jan 06 2019