A291339 Primes p such that p^3*q^3 + p^3 + q^3 is prime, where q is the next prime after p.
2, 3, 7, 19, 37, 47, 83, 89, 107, 137, 181, 251, 257, 349, 379, 569, 631, 653, 677, 691, 797, 823, 839, 863, 883, 919, 1009, 1021, 1223, 1229, 1361, 1423, 1571, 1609, 1831, 1873, 1907, 1993, 2053, 2113, 2207, 2239, 2293, 2309, 2579, 2833, 3137, 3319, 3593, 3673
Offset: 1
Keywords
Examples
a(2) = 3 is prime; 5 is the next prime: 3^3*5^3 + 3^3 + 5^3 = 27*125 + 27 + 125 = 3527 that is a prime. a(3) = 7 is prime; 11 is the next prime: 7^3*11^3 + 7^3 + 11^3 = 343*1331 + 343 + 1331 = 458207 that is a prime.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Programs
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Magma
[p: p in PrimesUpTo (5000) | IsPrime(p^3*q^3+p^3+q^3)];
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Maple
select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^3+nextprime(p)^3)]), [seq(p, p=1..10^4)]);
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Mathematica
Prime@Select[Range[1000], PrimeQ[Prime[#]^3*Prime[# + 1]^3 + Prime[#]^3 + Prime[# + 1]^3] &] Select[Partition[Prime[Range[600]],2,1],PrimeQ[Times@@(#^3)+Total[#^3]]&][[;;,1]] (* Harvey P. Dale, Apr 28 2025 *)
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PARI
is(n) = my(q=nextprime(n+1)); ispseudoprime(n^3*q^3+n^3+q^3) forprime(p=1, 3700, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 22 2017
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PARI
list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p3+q3), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017