A084289 Primes p such that the arithmetic mean of p and the next prime after p is a true prime power from A025475.

3, 7, 61, 79, 619, 1669, 4093, 822631, 1324783, 2411797, 2588869, 2778877, 3243589, 3636631, 3736477, 5527189, 6115717, 6405943, 8720191, 9005989, 12752029, 16056031, 16589317, 18087991, 21743551, 25230511, 29343871, 34586131, 37736431, 39150037, 40056229

Offset: 1

Author

Labos Elemer, May 26 2003

Keywords

• nonn

Examples

			n = prime(9750374) = 174689077, next prime = 174689101, mean = 174689089 = 13217^2, a prime power. The arithmetic mean of two consecutive primes is never prime, while between two consecutive primes, prime powers occur. These prime powers are in the middle of gap: p+d/2 = q^w. The prime power is most often square and very rarely occurs more than once (see A053706).
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A053706, A000961, A025475.

Programs

Mathematica

fi[x_] := FactorInteger[x] ff[x_] := Length[FactorInteger[x]] Do[s=(Prime[n]+Prime[n+1])/2; s1=ff[s]; If[Equal[s1,1],Print[{n,p=Prime[n],s,fi[s],s-p,s1}]], {n,1,10000000}]
Select[Partition[Prime[Range[25*10^5]],2,1],PrimePowerQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Oct 15 2023 *)

Formula

Primes p(j) such that (p(j)+p(j+1))/2 = q(m)^w, where q(m) is a prime.