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

Author

K. D. Bajpai, Aug 09 2014

Keywords

• nonn
• less

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A001358, A002327, A002496.

Programs

Magma

IsSemiprime:= func; [s: n in [1..400] | IsSemiprime(s) where s is n^4 - n^3 - n - 1]; // Vincenzo Librandi, Aug 10 2014

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 &]

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