A084704 Smallest prime p > prime(n) such that (p + prime(n))/2 is prime.

7, 17, 19, 23, 61, 29, 43, 59, 53, 43, 97, 53, 79, 59, 89, 83, 73, 79, 107, 181, 127, 131, 113, 109, 113, 151, 167, 193, 149, 151, 167, 197, 163, 197, 163, 229, 199, 179, 281, 347, 241, 263, 229, 257, 223, 271, 331, 239, 313, 269, 263, 313, 263, 269, 359, 293

Offset: 2

Author

Amarnath Murthy, Jun 08 2003

Keywords

• nonn

Comments

Subsidiary sequences: (1) Sequence of primes for a given prime p such that (p+q)/2 is a prime iff q belongs to this sequence. For example, for p = 5 the sequence is 17, 29, 41, 53, 89,...

Note that actually a(n) > prime(n+1) in all cases - because there is no prime between prime(n) and prime(n+1). - Zak Seidov, Jul 24 2013.

For n>2, a(n)-prime(n) is a multiple of 12. - Zak Seidov, Oct 15 2015

[Proof: the sequence searches prime triples prime(n)R. J. Mathar, Oct 16 2015]

Links

• T. D. Noe and Zak Seidov, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001358, A165138.

Programs

Maple

A084704 := proc(n)
    local p,a,q ;
    p := ithprime(n) ;
    a := nextprime(p) ;
    while not isprime((a+p)/2) do
        a := nextprime(a) ;
    end do:
    return a;
end proc: # R. J. Mathar, Oct 16 2015

Mathematica

Table[p = q = Prime[n]; While[q = NextPrime[q]; ! PrimeQ[(p + q)/2]]; q, {n, 2, 100}] (* T. D. Noe, Apr 20 2011 *)
p=2; Table[p=NextPrime[p]; q=NextPrime[p,2]; While[!PrimeQ[(p+q)/2], q=NextPrime[q]]; q, {99}] (* Zak Seidov, Jul 24 2013 *)

PARI

a(n) = {q = prime(n); p = nextprime(q+1); while (!isprime((q+p)/2), p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 15 2015

Extensions

More terms from David Wasserman, Jan 03 2005