A165138 -id:A165138 - cp's OEIS frontend

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

Amarnath Murthy, Jun 08 2003

nonn

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]

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

Cf. A001358, A165138.

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

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 *)

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

More terms from David Wasserman, Jan 03 2005

A224895 Let p = prime(n). Smallest odd number m > p such that m + p is semiprime.

Original entry on oeis.org

7, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273

Offset: 1

Zak Seidov, Jul 24 2013

nonn

Apparently a(n) = A210497(n) for n>1, which basically indicates that the search for the smallest even semiprime larger than 2*prime(n) produces 2*prime(n+1). - R. J. Mathar, Jul 27 2013
a(n) <= A165138(n); a(n) = A165138(n) when a(n) is prime, corresponding n's: 1, 2, 11, 15, 18, 36, 39, 46, 54, 55, 58, 73, 91,.. .
Also of interest is that sequence in not monotonic: e.g., a(10) - a(9) = 33 - 35 = -2, a(31) - a(30) = 135 - 141 = -6.

			2 + 7 = 9 = 3*3, 3 + 7 = 10 = 2*5, 5 + 9 = 14 = 2*7.

Cf. A000040, A001358, A084704, A165138, A211280, A210497.

A224895 := proc(n)
    local p,m ;
    p := ithprime(n) ;
    for m from p+1 do
        if type(m,'odd') and numtheory[bigomega](m+p) = 2 then
            return m ;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 28 2013

Reap[Sow[7];Do[p=Prime[n];k=p+2;While[!PrimeQ[r=(p+k)/2],k=k+2];Sow[k],{n,2,100}]][[2,1]]
son[n_]:=Module[{m=If[EvenQ[n],n+1,n+2]},While[PrimeOmega[n+m]!=2,m = m+2]; m]; Table[son[n],{n,Prime[Range[60]]}] (* Harvey P. Dale, Apr 24 2017 *)

cp's OEIS Frontend

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

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