A321891 Prime numbers of the form p^3 + q, where p and q are primes.

11, 13, 19, 29, 31, 37, 61, 67, 79, 97, 109, 127, 139, 157, 181, 199, 241, 271, 277, 367, 397, 409, 439, 457, 487, 499, 571, 577, 601, 607, 661, 691, 709, 727, 751, 769, 829, 919, 937, 991, 1021, 1039, 1069, 1117, 1171, 1201, 1231, 1237, 1291, 1297, 1327, 1381

Offset: 1

Author

Pierandrea Formusa, Nov 20 2018

Keywords

• nonn

Comments

For reasons of parity, either p or q must be equal to 2, so this actually is the union of (mostly) "primes of the form p + 8" (A092402) and (rarely) "primes of the form p^3 + 2" (A048636 = 29, 127, 24391, 357913, ...). - M. F. Hasler, Jan 13 2025

Except for 13, these primes are the minimum or maximum prime numbers of the respective decade. - Davide Rotondo, Jan 31 2025

Examples

			37 is prime and 37 = 2^3 + 29, where 2 and 29 are primes, therefore 37 is a term.
		

Links

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

Crossrefs

Union of A048636 and A092402. - Michel Marcus, Nov 21 2018

Programs

Maple

N:= 2000: # to get terms <= N
A1:= select(t -> isprime(t) and isprime(t-8), {11,seq(i,i=13 ..N,6)}):
v:= floor((N-2)^(1/3)):
B:= select(t -> isprime(t) and isprime(t^3+2), {3,seq(i,i=5..v,6)}):
sort(convert(A1 union map(t -> t^3+2,B), list)); # Robert Israel, Mar 05 2020

Mathematica

nmax=4; Select[Union[Prime[Range[nmax]]^3 + 2, Prime[Range[Prime[nmax]^3]] + 8], PrimeQ] (* Amiram Eldar, Nov 21 2018 *)

MiniZinc

include "globals.mzn";
int: n = 2;
int: max_val = 1200000;
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 /\
pow(x[1], 3)+pow(x[2], 1)= x[3] ;
output [ show(x)]

PARI

list(lim)=my(v=List()); forprime(p=3,sqrtnint((lim\=1)-2,3), if(isprime(p^3+2), listput(v,p^3+2))); forprime(p=11,lim+8, if(isprime(p-8), listput(v,p))); Set(v) \\ Charles R Greathouse IV, Jan 13 2025

PARI

select( {is_A321891(n)=isprime(n)&& (isprime(n-8)|| (ispower(n-2, 3, &n)&&isprime(n)))}, [1..1234]) \\ M. F. Hasler, Jan 13 2025

Extensions

More terms from Amiram Eldar, Nov 21 2018