A383504 Sum of next a(n) successive prime squares is prime.

2, 5, 7, 25, 11, 13, 7, 7, 35, 7, 19, 31, 25, 11, 5, 19, 5, 11, 5, 139, 37, 17, 19, 19, 13, 5, 7, 7, 19, 13, 11, 5, 7, 11, 5, 5, 5, 13, 47, 43, 5, 23, 13, 11, 11, 79, 31, 35, 5, 5, 25, 5, 37, 95, 31, 13, 43, 17, 5, 35, 17, 23, 11, 41, 59, 7, 47, 5, 13, 7, 11

Offset: 1

Author

Abhiram R Devesh, May 18 2025

Keywords

• nonn

Comments

Group the primes such that the sum of squares of members of each group is a prime, and each successive group is as short as possible.

Apart from a(1)=2, a(n) is odd and not a multiple of 3.

Examples

			Primes, their squares, and the lengths of blocks which sum to a prime begin,
  primes  2, 3,  5,  7,  11,  13,  17,  19, ...
  squared 4, 9, 25, 49, 121, 169, 289, 361, ...
          \--/  \-------------------/  \--- ...
  sum      13            653
  a(n) =    2             5
		

Links

• Abhiram R Devesh, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001248, A073684 (sum of successive primes), A384161 (sum of successive prime cubes).

Programs

Maple

i:= 0: p:= 0: t:= 0: count:= 0: R:= NULL:
while count < 100 do
  p:= nextprime(p);
  i:= i+1;
  t:= t + p^2;
  if isprime(t) then
    R:= R, i; count:= count+1; i:= 0; t:= 0;
  fi
od:
R; # Robert Israel, May 25 2025

Mathematica

p=1;s={};Do[c=0;sm=0;While[!PrimeQ[sm],sm=sm+Prime[p]^2;p++;c++];AppendTo[s,c],{n,71}];s (* James C. McMahon, Jun 09 2025 *)

Python

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s, i, p = 0, 1, 2
    while True:
        while not(isprime(s:=s+p**2)): i, p = i+1, nextprime(p)
        yield i
        s, i, p = 0, 1, nextprime(p)
print(list(islice(agen(), 71))) # Michael S. Branicky, May 23 2025