A180481 The smallest prime q > p = prime(n) such that p*(q-p)+q, p*(q-p)-q, q*(q-p)+p and q*(q-p)-p are simultaneously prime, or 0 if no such q exists.

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

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*(q-p)+q is divisible by 3. A similar reasoning applies for p = 6k - 1: here q - p = 6n + 4 entails 3|q, and q - p = 6n + 2 yields 3 | p*(q-p)-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.

Mathematica

sp[p_]:=Module[{q=NextPrime[p]}, While[AnyTrue[{p(q-p)+q, p(q-p)-q, q(q-p)+p, q(q-p)-p}, CompositeQ], q=NextPrime[q]]; q]; Join[{11, 23}, Table[sp[p], {p, Prime[Range[3, 60]]}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 23 2021 *)

Prog

(PARI) A180481(p)={ forprime( q=1+p=prime(p), default(primelimit), isprime(p*(q-p)+q)|next; isprime(p*(q-p)-q)|next; isprime(q*(q-p)+p)|next; isprime(q*(q-p)-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*(q-p)+q) and isprime(p*(q-p)-q) and isprime(q*(q-p)+p) and isprime(q*(q-p)-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: A351849 A253684 A364165 * A110044 A032663 A352868

Adjacent sequences: A180478 A180479 A180480 * A180482 A180483 A180484

Keyword

nonn

Author

M. F. Hasler, Jan 20 2011

Status

approved