A316971 Prime numbers of the form p1^3 + p2^2 + p3, with p1, p2 and p3 also prime and all different.

59, 71, 83, 89, 107, 113, 131, 137, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 223, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419

Offset

1,1

Comments

Conjecture: the only primes not in the sequence are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 73, 79, 97, 101, 103, 109, 127, 139, 199, 229, 463. - Robert Israel, Aug 17 2018

Links

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

Example

59 belongs to this sequence as 59 = 3^3 + 5^2 + 7, with 3, 5 and 7 all different primes.

Maple

N:= 500: # to get all terms <= N
p1:= 2: Res:= {}:
do
  p1:= nextprime(p1);
  if p1^3 + 3^2+5 > N then break fi;
  p2:= 2;
  do
    p2:= nextprime(p2);
    if p2 = p1 then next fi;
    if p1^3 + p2^2 + 3 > N then break fi;
    p3:= 2;
    do
      p3:= nextprime(p3);
      if p3=p1 or p3=p2 then next fi;
      v:= p1^3 + p2^2 + p3;
      if v > N then break fi;
      if isprime(v) then Res:= Res union {v} fi
od od od:
sort(convert(Res, list)); # Robert Israel, Aug 17 2018

Mathematica

v[t_] := Prime@Range@PrimePi@t; up=500; Union@ Reap[Do[If[ PrimeQ[p = p1^3 + p2^2 + p3] && p1!=p2 && p2!=p3 && p3!=p1, Sow@p], {p1, v[up^(1/3)]}, {p2, v@Sqrt[up - p1^3]}, {p3, v[up - p1^3 - p2^2]}]][[2, 1]] (* Giovanni Resta, Jul 18 2018 *)

Prog

(Minizinc)
%Model to get all primes less than 300 of this sequence
include "globals.mzn";
int: n = 3;
int: max_val = 300;
array[1..n+1] of var 2..max_val: x;
% primes between 2..max_val
set of int: prime = 2..max_val diff { i | i in 2..max_val, j in 2..ceil(sqrt(i)) where i mod j = 0} ;
set of int: primes; primes = prime union {2};
solve satisfy;
constraint all_different(x) /\ x[1] in primes /\ x[2] in primes /\ x[3] in primes /\ x[4] in primes /\ pow(x[1], 3)+pow(x[2], 2)+pow(x[3], 1) = x[4];
output [ show(x[4]) ];

Crossrefs

Cf. A000040.

Sequence in context: A065208 A061759 A061764 * A162000 A080192 A126693

Adjacent sequences: A316968 A316969 A316970 * A316972 A316973 A316974

Keyword

nonn

Author

Pierandrea Formusa, Jul 17 2018

Status

approved