A263675 Numbers that are both averages of consecutive primes and nontrivial prime powers.

4, 9, 64, 81, 625, 1681, 4096, 822649, 1324801, 2411809, 2588881, 2778889, 3243601, 3636649, 3736489, 5527201, 6115729, 6405961, 8720209, 9006001, 12752041, 16056049, 16589329, 18088009, 21743569, 25230529, 29343889, 34586161, 37736449, 39150049

Offset: 1

Author

Antonio Roldán, Oct 23 2015

Keywords

• nonn

Comments

Intersection of A024675 and A025475.

Lesser of consecutive primes is in the sequence A084289.

Examples

			625 is in this sequence because 625 = 5^4, nontrivial prime power, and 625 = (619+631)/2, with 619 and 631 consecutive primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1560
• Eric Weisstein's World of Mathematics, Interprime

Crossrefs

Cf. A075190, A075277, A075296, A078443, A084289, A130178, A263674, A263676.

Programs

Maple

N:= 10^10: # to get all terms <= N
Primes:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
S:= select(t -> t - prevprime(t) = nextprime(t)-t, {seq(seq(p^j, j=2..floor(log[p](N))),p=Primes)}):
sort(convert(S,list)); # Robert Israel, Dec 27 2015

Mathematica

(* version >= 6 *)(#/2 + NextPrime[#]/2) & /@
Select[Prime[Range[5000000]], PrimePowerQ[#/2 + NextPrime[#]/2] &]
(* Wouter Meeussen, Oct 26 2015 *)

PARI

{for(i=1,10^8,if(isprimepower(i)>1&&i==(precprime(i-1)+nextprime(i+1))/2,print1(i,", ")))}