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A318158
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Prime numbers of the form p1^4 + p2^3 + p3^2 + p4, where p1, p2, p3 and p4 are distinct primes.
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1
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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
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OFFSET
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1,1
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COMMENTS
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Does this sequence contain every prime > 113? - Robert Israel, Aug 26 2018
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)
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LINKS
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EXAMPLE
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227 belongs to this sequence as 227 = 3^4 + 5^3 + 2^2 + 17, with 2, 3, 5 and 17 all primes.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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])]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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