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

1,1

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A053706, A000961, A025475.

Sequence in context: A046859 A242380 A225195 * A183174 A329966 A255669

Adjacent sequences: A084286 A084287 A084288 * A084290 A084291 A084292

Keyword

nonn

Author

Labos Elemer, May 26 2003

Status

approved