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

1,1

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A106018 A188499 A215535 * A056921 A041245 A042159

Adjacent sequences: A383501 A383502 A383503 * A383505 A383506 A383507

Keyword

nonn

Author

Abhiram R Devesh, May 18 2025

Status

approved