A105041 Positive integers k such that k^7 + 1 is semiprime.

2, 10, 16, 18, 46, 52, 66, 72, 78, 106, 136, 148, 226, 228, 240, 262, 282, 330, 442, 508, 616, 630, 732, 750, 756, 768, 810, 828, 910, 936, 982, 1032, 1060, 1128, 1216, 1302, 1366, 1558, 1626, 1696, 1698, 1758, 1800, 1810, 1830, 1932, 1996, 2002, 2026, 2080

Offset

1,1

Comments

We have the polynomial factorization n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1). Hence after the initial n=1 prime, the binomial can at best be semiprime and that only when both (n+1) and (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) are primes.

Links

Robert Price, Table of n, a(n) for n = 1..1414

Formula

a(n)^7 + 1 is semiprime. a(n)+1 is prime and a(n)^6 - a(n)^5 + a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.

Example

n n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
2 129 = 3 * 43
10 10000001 = 11 * 909091
16 268435457 = 17 * 15790321
18 612220033 = 19 * 32222107
46 435817657217 = 47 * 9272716111

Mathematica

Select[Range[0, 200000], PrimeQ[# + 1] && PrimeQ[(#^7 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)
Select[Range[2500], Plus@@Last/@FactorInteger[#^7 + 1]==2 &] (* Vincenzo Librandi, Mar 12 2015 *)
Select[Range[2100], PrimeOmega[#^7+1]==2&] (* Harvey P. Dale, Jun 18 2019 *)

Prog

(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..2100] | IsSemiprime(n^7+1)]; // Vincenzo Librandi, Mar 12 2015
(PARI) is(n)=isprime(n+1) && isprime((n^7+1)/(n+1)) \\ Charles R Greathouse IV, Aug 31 2021

Crossrefs

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

Sequence in context: A048043 A370115 A043429 * A138632 A175957 A307055

Adjacent sequences: A105038 A105039 A105040 * A105042 A105043 A105044

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 03 2005

Extensions

More terms from R. J. Mathar, Dec 14 2009

Status

approved