A105122 Positive integers n such that n^11 + 1 is semiprime.

2, 6, 12, 232, 262, 280, 330, 430, 508, 772, 786, 852, 1012, 1522, 1566, 1626, 1810, 2346, 2556, 2676, 3658, 3888, 3910, 4018, 4048, 4258, 4830, 5188, 5322, 5478, 5848, 6090, 6366, 6568, 7018, 7458, 7602, 7606, 7822, 8178, 8928, 9420, 9618, 9676, 10398

Offset

1,1

Comments

We have the polynomial factorization n^11+1 = (n+1) * (n^10 - n^9 + n^8 - n^7 + 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^10 - n^9 + n^8 - n^7 + 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..4303

Formula

a(n)^11+1 is semiprime A001538. a(n)+1 is prime and a(n)^10 - a(n)^9 + a(n)^8 - a(n)^7 + a(n)^6 - a(n)^5 + a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.

Example

2^11+1 = 2049 = 3 * 683,
6^11+1 = 362797057 = 7 * 51828151,
1012^11+1 = 1140212079231804336089593374834689 = 1013 * 1125579545144920371263172137053.

Mathematica

Select[ Range[10721], PrimeQ[ # + 1] && PrimeQ[(#^11 + 1)/(# + 1)] &] (* Robert G. Wilson v, Apr 09 2005 *)

Crossrefs

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

Sequence in context: A319481 A195338 A179201 * A351940 A132076 A309743

Adjacent sequences: A105119 A105120 A105121 * A105123 A105124 A105125

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 08 2005

Extensions

More terms from Robert G. Wilson v, Apr 09 2005

Status

approved