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 A084289 Primes p such that the arithmetic mean of p and the next prime after p is a true prime power from A025475. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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). MAPLE 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}] CROSSREFS Cf. A053706, A000961, A025475. Sequence in context: A046859 A242380 A225195 * A183174 A255669 A258184 Adjacent sequences:  A084286 A084287 A084288 * A084290 A084291 A084292 KEYWORD nonn AUTHOR Labos Elemer, May 26 2003 STATUS approved

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Last modified October 21 04:40 EDT 2019. Contains 328291 sequences. (Running on oeis4.)