A103854 Positive integers n such that n^6 + 1 is semiprime.

2, 4, 10, 36, 56, 94, 126, 224, 260, 270, 300, 350, 686, 716, 780, 1036, 1070, 1080, 1156, 1174, 1210, 1394, 1416, 1434, 1440, 1460, 1524, 1550, 1576, 1616, 1654, 1660, 1700, 1756, 1860, 1980, 2054, 2084, 2096, 2116, 2224, 2454, 2600, 2664, 2770, 2864

Offset: 1

Author

Jonathan Vos Post, Mar 31 2005

Keywords

• easy
• nonn

Comments

n^6+1 can only be prime when n = 1, n^6+1 = 2. This is because the sum of cubes formula gives the polynomial factorization n^6+1 = (n^2+1) * (n^4 - n^2 + 1). Hence n^6+1 can only be semiprime when both (n^2+1) and (n^4 - n^2 + 1) are primes.

Examples

			n n^6+1 = (n^2+1) * (n^4 - n^2 + 1)
2 65 = 5 * 13
4 4097 = 17 * 241
10 1000001 = 101 * 9901
36 2176782337 = 1297 * 1678321
56 30840979457 = 3137 * 9831361
94 689869781057 = 8837 * 78066061
126 4001504141377 = 15877 * 252031501
224 126324651851777 = 50177 * 2517580801
		

Links

• Robert Price, Table of n, a(n) for n = 1..1134

Crossrefs

Subsequence of A005574.

Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Programs

Mathematica

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ 2Range@1526, semiprimeQ[ #^6 + 1] &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[200000], PrimeQ[#^2 + 1] && PrimeQ[(#^6 + 1)/(#^2 + 1)] &] (* Robert Price, Mar 11 2015 *)

PARI

is(n)=my(s=n^2); isprime(s+1) && isprime(s^2-s+1) \\ Charles R Greathouse IV, Aug 31 2021

Formula

a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.

Extensions

More terms from Robert G. Wilson v, May 26 2006