

A180481


The smallest prime q > p = prime(n) such that p*(qp)+q, p*(qp)q, q*(qp)+p and q*(qp)p are simultaneously prime, or 0 if no such q exists.


2



11, 23, 11, 67, 3119, 19, 941, 739, 29, 41, 79, 127, 5507, 1399, 191, 56873, 1193, 16657, 49411, 30059, 10453, 373, 719, 18773, 12277, 1031, 1489, 131, 823, 1283, 14251, 317, 10631, 313, 191, 16987, 70381, 229, 8447, 3539, 1019, 3499, 2777, 301579, 587, 241, 6229, 229, 11657, 571, 2969, 701, 1627, 20327, 467, 2069, 863
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OFFSET

1,1


COMMENTS

It is conjectured that a(n) > 0 for all n, and for infinitely many terms, a(n) = prime(n+1).
a(n) = prime(n+1) for n = 9, 100, 508, 627, 752, 835, 889, ... (that is, for p = 23, 541, 3631, 4643, 5711, 6421, 6911, ...)  Derek Orr, Aug 25 2014
We have a(n)  prime(n) == 0 (mod 6) for all n > 2. Indeed, suppose p = 6k + 1, then q  p = 6n + 2 would imply that q is divisible by 3, and q  p = 6n + 4 would imply that p*(qp)+q is divisible by 3. A similar reasoning applies for p = 6k  1: here q  p = 6n + 4 entails 3q, and q  p = 6n + 2 yields 3  p*(qp)q.


LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000
W. Sindelar, Certain Pairs of Consecutive Prime Numbers, in yahoo group "primenumbers", Jan 20 2011.
W. Sindelar, David Broadhurst, Certain Pairs of Consecutive Prime Numbers, digest of 2 messages in primenumbers Yahoo group, Jan 20  Jan 21, 2011.


FORMULA

a(n) = A000040(n) + 6*A180476(n) for all n > 2.


PROG

(PARI) A180481(p)={ forprime( q=1+p=prime(p), default(primelimit), isprime(p*(qp)+q)next; isprime(p*(qp)q)next; isprime(q*(qp)+p)next; isprime(q*(qp)p)next; return(q)) }
(Python)
from sympy import prime, isprime
def A180481(n):
....p = prime(n)
....n += 1
....q = prime(n)
....while q < 10**14: # note: search limit
........if isprime(p*(qp)+q) and isprime(p*(qp)q) and isprime(q*(qp)+p) and isprime(q*(qp)p):
............return(q)
........n += 1
........q = prime(n)
....return(0) # limit in search for q was reached. A180481(n) may be > 0
# Chai Wah Wu, Aug 24 2014


CROSSREFS

Cf. A180476.
Sequence in context: A225186 A155973 A253684 * A110044 A032663 A119815
Adjacent sequences: A180478 A180479 A180480 * A180482 A180483 A180484


KEYWORD

nonn


AUTHOR

M. F. Hasler, Jan 20 2011


STATUS

approved



