A380319 Smallest prime that is the sum of 2n+1 squares of consecutive odd primes, or 0 if no such prime exists.

83, 373, 1543, 2393, 4723, 10453, 0, 24953, 35323, 0, 56383, 98017, 0, 122701, 238879, 0, 263723, 318181, 0, 486617, 816547, 0, 874487, 817561, 0, 1130957, 1203343, 0, 3110867, 2451637, 1789391, 1987849, 2331379, 0, 2706679, 3124129, 0, 4260437, 4446319, 0

Offset

1,1

Comments

If n == 1 (mod 3), either a(n) = 0 or a(n) is the sum of the squares of the primes 3, 5, ..., prime(2*n+2), because p^2 == 1 (mod 3) for all primes p > 3 so the sum of 2*n+1 such primes is divisible by 3. - Pontus von Brömssen, Jan 20 2025

If n == 1 (mod 3), a(n) > 0 iff n is in A370633. - Robert Israel, Nov 27 2025

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

P:= map(`^`, select(isprime, [seq(i, i=3..10^7, 2)]), 2):
PS:= ListTools:-PartialSums([0, op(P)]):
N:= nops(PS):
f:= proc(n) local k;
  if n mod 3 = 1 then
     if isprime(PS[2*n+2]) then return PS[2*n+2] else return 0 fi
  fi;
  for k from 1 to N-(2*n+1) do
    if isprime(PS[k+2*n+1]-PS[k]) then return PS[k+2*n+1]-PS[k] fi
  od;
  FAIL
end proc:
map(f, [$1..100]); # Robert Israel, Nov 27 2025

Crossrefs

Cf. A001248 (primes squared), A082244 (analog for primes), A379760 (analog for cube of primes).

Cf. A370633, A372041.

Sequence in context: A327850 A061525 A293979 * A340861 A160359 A142496

Adjacent sequences: A380316 A380317 A380318 * A380320 A380321 A380322

Keyword

nonn

Author

Michel Marcus, Jan 20 2025

Extensions

Escape clause and more terms added by Pontus von Brömssen, Jan 20 2025

Status

approved