A291339 -id:A291339 - cp's OEIS frontend

A291374 Primes p such that p^3*q^3 + p + q is prime, where q is next prime after p.

11, 17, 41, 43, 47, 137, 313, 359, 389, 401, 491, 557, 577, 709, 757, 829, 863, 929, 937, 953, 1129, 1163, 1249, 1301, 1307, 1439, 1597, 1627, 1693, 1847, 2087, 2311, 2351, 2437, 2663, 2731, 2741, 3109, 3119, 3217, 3253, 4027, 4219, 4271, 4547, 4637, 5189, 5237

Offset: 1

K. D. Bajpai, Aug 23 2017

nonn

			a(1) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11 + 13 = 1331*2197 + 11 + 13 = 2924231 that is a prime.
a(2) = 17 is prime; 19 is the next prime: 17^3*19^3 + 17 + 19 = 4913*6859 + 17 + 19 = 33698303 that is a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339.

[p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p + q) where q is NextPrime(p)];

select(p -> andmap(isprime, [p,(p^3*nextprime(p)^3+p+nextprime(p))]), [seq(p,p=1..10^4)]);

Prime@Select[Range[1000], PrimeQ[Prime[#]^3* Prime[# + 1]^3 + Prime[#] + Prime[# + 1]] &]

forprime(p=1,5000, q=nextprime(p+1); if(ispseudoprime(p^3*q^3 + p + q), print1(p, ", ")));

list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p+q), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017

A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

Original entry on oeis.org

2, 11, 13, 41, 97, 277, 389, 1093, 1229, 1409, 1429, 1627, 1823, 1931, 1979, 2437, 2521, 2549, 2657, 2689, 2719, 2729, 2731, 2969, 3019, 3413, 3539, 3593, 3613, 3623, 3697, 4003, 4027, 4289, 4327, 4583, 4751, 5051, 5323, 5503, 5657, 5783, 6143, 6221, 6299, 6329

Offset: 1

K. D. Bajpai, Aug 24 2017

nonn

			a(1) = 2 is prime; 3 is the next prime: 2^3*3^3 + 2^2 + 3^2 = 8*27 + 4 + 9 = 229 that is a prime.
a(2) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11^2 + 13^2 = 1331*2197 + 121 + 169 = 2924497 that is a prime.

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

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339, A291374.

[p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p^2 + q^2) where q is NextPrime(p)];

select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^2+nextprime(p)^2)]), [seq(p, p=1..10^4)]);

Select[Prime[Range[5000]], PrimeQ[#^3*NextPrime[#]^3 + #^2 + NextPrime[#]^2] &]
Select[Partition[Prime[Range[1000]],2,1],PrimeQ[#[[1]]^3 #[[2]]^3+#[[1]]^2+#[[2]]^2]&][[;;,1]] (* Harvey P. Dale, Sep 11 2023 *)

forprime(p=1, 5000, q=nextprime(p+1); p3=p^3; p2=p^2; q3=q^3; q2=q^2; if(ispseudoprime(p3*q3 + p2 + q2), print1(p, ", ")));

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A291374 Primes p such that p^3*q^3 + p + q is prime, where q is next prime after p.

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A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

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Programs

Magma

Maple

Mathematica

PARI