A384161 Sum of next a(n) successive prime cubes is prime.

4, 7, 3, 11, 13, 9, 131, 9, 15, 3, 31, 27, 3, 13, 7, 3, 31, 131, 15, 17, 13, 5, 21, 29, 3, 33, 3, 7, 11, 43, 5, 41, 43, 49, 27, 49, 37, 85, 5, 41, 3, 41, 65, 51, 13, 29, 65, 5, 89, 3, 27, 75, 3, 73, 3, 3, 5, 3, 23, 9, 7, 3, 71, 55, 35, 7, 71, 71, 19, 33, 15

Offset

1,1

Comments

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

Links

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

Example

Primes, their cubes and the lengths of the blocks when summed becomes a prime.
Primes 2,  3,   5,   7,   11,   13,   17,   19,    23,    29,    31,    37,    41
Cubes  8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921
      \--------------/  \------------------------------------------/ \---...
Sum         503                           81647
a(n) =       4                              7

Maple

i:= 0: p:= 0: t:= 0: count:= 0: R:= NULL:
while count < 100 do
  p:= nextprime(p);
  i:= i+1;
  t:= t + p^3;
  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]^3; 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**3)): 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. A030078, A073684 (sum of successive primes), A383504 (sum of successive prime squares).

Sequence in context: A130204 A021215 A063378 * A280547 A301930 A365940

Adjacent sequences: A384158 A384159 A384160 * A384162 A384163 A384164

Keyword

nonn

Author

Abhiram R Devesh, May 20 2025

Status

approved