A318158 Prime numbers of the form p1^4 + p2^3 + p3^2 + p4, where p1, p2, p3 and p4 are distinct primes.

79, 97, 103, 109, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 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

Offset

1,1

Comments

Does this sequence contain every prime > 113? - Robert Israel, Aug 26 2018

From David A. Corneth, Aug 26 2018: (Start)

As the primes in the sum are distinct and has four terms, exactly one of (p1, p2, p3, p4) is 2.

Contains all the primes in [120, 5 * 10^7]. (End)

Links

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

David A. Corneth, Statistic for computations using the PARI program below

David A. Corneth, PARI program

Example

227 belongs to this sequence as 227 = 3^4 + 5^3 + 2^2 + 17, with 2, 3, 5 and 17 all primes.

Maple

N:= 1000: # to get all terms <= N
V:= Vector(N):
p1:= 1:
do
  p1:= nextprime(p1);
  if p1^4 > N then break fi;
  p2:= 1:
  do
    p2:= nextprime(p2);
    if p1^4 + p2^3 > N then break fi;
    if p2 = p1 then next fi;
    p3:= 1;
    do
      p3:= nextprime(p3);
      if p1^4 + p2^3 + p3^2 > N then break fi;
      if p3 = p1 or p3 = p2 then next fi;
      if min(p1, p2, p3)>2 then
         p4:= 2;
         x:= p1^4+p2^3+p3^2+p4;
         if isprime(x) then V[x]:= 1 fi;
      else
         p4:= 2;
         do
            p4:= nextprime(p4);
            if p1^4 + p2^3 + p3^2 + p4 > N then break fi;
            if p4 = p1 or p4 = p2 or p4 = p3 then next fi;
            x:= p1^4+p2^3+p3^2+p4;
            if isprime(x) then V[x]:= 1 fi;
          od
       fi
od od od:
select(t -> V[t]=1, [$1..N]); # Robert Israel, Aug 26 2018

Mathematica

v[t_] := Prime@Range@PrimePi@t; up = 400; Union@Reap[ Do[ If[PrimeQ[p = p1^4 + p2^3 + p3^2 + p4] && (s = {p1, p2, p3, p4}; Sort@s == Union@s), Sow@p], {p1, v[ up^(1/4)]}, {p2, v@Sqrt[up - p1^4]}, {p3, v[up - p1^4 - p2^3]}, {p4, v[up - p1^4 - p2^3 - p3^2]}]][[2, 1]] (* Giovanni Resta, Aug 19 2018 *)

Prog

(MiniZinc)
include "globals.mzn";
int: n = 4;
%to get all primes less than 250 of this sequence
int: max_val = 250;
array[1..n+1] of var 2..max_val: x;
% primes between 2..max_valset 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 /\
x[5] in primes /\
pow(x[4], 4)+pow(x[3], 3)+pow(x[2], 2)+pow(x[1], 1)= x[5] ;
output [ show(x[5])]

Crossrefs

Cf. A316971.

Sequence in context: A235227 A039544 A091819 * A117244 A039436 A043259

Adjacent sequences: A318155 A318156 A318157 * A318159 A318160 A318161

Keyword

nonn

Author

Pierandrea Formusa, Aug 19 2018

Extensions

More terms from Giovanni Resta, Aug 19 2018

Status

approved