A359322 a(n) is the first prime p such that the average of the squares of n consecutive primes starting with p is prime.

3, 7, 7, 1627, 83, 7, 23, 7, 19, 17, 73, 281, 179, 257, 5, 43, 73, 43, 19, 67, 911, 193, 7, 1613, 139, 383, 7, 719, 113, 967, 31, 19, 211, 769, 149, 173, 13, 13, 59, 137, 23, 47, 89, 607, 61, 127, 61, 317, 1049, 1277, 547, 281, 317, 4157, 199, 107, 373, 149, 229, 367, 1489, 643, 563, 587, 263

Offset

2,1

Comments

Suggested in an email by J. M. Bergot.

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Example

a(3) = 7 because 7, 11, 13 are 3 consecutive primes and (7^2 + 11^2 + 13^2)/3 = 113 is prime, and 7 is the least prime that works.

Maple

f:= proc(n) local P, s, i;
   P:= <seq(ithprime(i), i=1..n)>;
   s:= add(P[i]^2, i=1..n)/n;
   do
     if s::integer and isprime(s) then return P[1] fi;
     s:= s - P[1]^2/n;
     P[1..-2] := P[2..-1]; P[n]:= nextprime(P[n]);
     s:= s + P[n]^2/n;
   od
end proc:
map(f, [$2..70]);

Prog

(Python)
from sympy import prime, isprime, nextprime, primerange
def a(n):
    plst = list(primerange(2, prime(n)+1))
    powsum = sum(p**2 for p in plst)
    while True:
        q, r = divmod(powsum, n)
        if r == 0 and isprime(q): return plst[0]
        powsum -= plst[0]**2
        plst = plst[1:] + [nextprime(plst[-1])]
        powsum += plst[-1]**2
print([a(n) for n in range(2, 67)]) # Michael S. Branicky, Dec 29 2022

Crossrefs

Sequence in context: A336719 A272981 A086839 * A316258 A258405 A064208

Adjacent sequences: A359319 A359320 A359321 * A359323 A359324 A359325

Keyword

nonn

Author

Robert Israel, Dec 25 2022

Status

approved