A241670 Semiprimes of the form n^4 - n^3 - n - 1.

187, 1073, 8989, 35657, 61423, 151979, 1632923, 2495959, 8345537, 9658823, 18687173, 49194347, 64880909, 77244217, 179502923, 250046873, 451259573, 502874849, 588444323, 651263839, 830296829, 1723401587, 1935548789, 4552183739, 4839132407, 8739047573, 13324055659

Offset

1,1

Comments

Since n^4 - n^3 - n - 1 = (n^2 + 1)*(n^2 - n - 1), it is a must that (n^2 + 1) and (n^2 - n - 1) both should be prime.

Primes of the form (n^2+1) are at A002496.

Primes of the form (n^2-n-1) are at A002327.

Links

K. D. Bajpai, Table of n, a(n) for n = 1..5330

Example

187 is in the sequence because 4^4 - 4^3 - 4 - 1 = 187 = 11 * 17, which is semiprime.
1073 is in the sequence because 6^4 - 6^3 - 6 - 1 = 1073 = 29 * 37, which is semiprime.

Maple

select(k -> numtheory:-bigomega(k)=2, [seq(x^4 - x^3 - x - 1, x=1..1000)]);

Mathematica

Select[Table[n^4 - n^3 - n - 1, {n, 500}], PrimeOmega[#] == 2 &]

Prog

(PARI)
for(n=1, 10^4, p=n^2+1; q=n^2-n-1; if(isprime(p)&&isprime(q), print1(p*q, ", "))) \\ Derek Orr, Aug 09 2014
(Magma) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..400] | IsSemiprime(s) where s is n^4 - n^3 - n - 1]; // Vincenzo Librandi, Aug 10 2014

Crossrefs

Cf. A000040, A001358, A002327, A002496.

Sequence in context: A308810 A063346 A350044 * A134163 A030536 A222911

Adjacent sequences: A241667 A241668 A241669 * A241671 A241672 A241673

Keyword

nonn,less

Author

K. D. Bajpai, Aug 09 2014

Status

approved