

A038771


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


2



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
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OFFSET

0,1


COMMENTS

The lower "envelope" of the sequence is prime(n+1)^2. See also Fortuneconjecture (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


PROG

(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


CROSSREFS

Cf. A002110, A054757, A054758, A005235.
Sequence in context: A188836 A030146 A229970 * A158146 A158147 A158148
Adjacent sequences: A038768 A038769 A038770 * A038772 A038773 A038774


KEYWORD

nonn


AUTHOR

Labos Elemer, May 04 2000


EXTENSIONS

Name edited by Tom Edgar, Jun 08 2015
a(0) prepended by Dmitry Kamenetsky, Jan 06 2019


STATUS

approved



