A357133 a(n) is the least prime that is the arithmetic mean of n consecutive primes.

5, 127, 79, 101, 17, 269, 491, 727, 53, 23, 71, 181, 29, 31, 37, 43, 563, 331, 883, 283, 173, 307, 157, 113, 353, 571, 347, 89, 263, 139, 179, 1201, 281, 1553, 137, 5167, 347, 563, 2083, 2087, 491, 1867, 353, 463, 1973, 199, 599, 4373, 149, 9929, 277, 463, 1259, 251, 397, 2897, 787, 263, 2161

Offset: 3

Author

J. M. Bergot and Robert Israel, Sep 14 2022

Keywords

• nonn

Examples

			a(5) = 79 because 79 is the average of the 5 consecutive primes 71, 73, 79, 83, 89, and 79 is the least prime that works.
		

Links

• Robert Israel, Table of n, a(n) for n = 3..7000

Crossrefs

Cf. A006562, A126096, A219478, A122531

Programs

Maple

P:= [seq(ithprime(i),i=1..10^5)]:
S:= ListTools:-PartialSums(P):
g:= proc(n) local k,r;
   for k from 1 to 10^5-n do
      r:= (S[k+n]-S[k])/n;
      if r::integer and isprime(r) then return r fi
   od;
   -1
end proc:
map(g, [$3..100]);

Mathematica

a={}; n=3; For[k=1, k<=1200, k++, If[PrimeQ[p=Sum[Prime[k+i], {i,0,n-1}]/n], AppendTo[a,p]; n++; k=1]]; a (* Stefano Spezia, Sep 15 2022 *)

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    n, plst0 = 3, [2, 3, 5]
    while True:
        plst = plst0[:]
        while True:
            q, r = divmod(sum(plst), n)
            if r == 0 and isprime(q): yield q; break
            plst = plst[1:] + [nextprime(plst[-1])]
        plst0.append(nextprime(plst0[-1]))
        n += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Sep 14 2022