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%I #25 Sep 08 2022 08:45:17
%S 2,10,16,18,46,52,66,72,78,106,136,148,226,228,240,262,282,330,442,
%T 508,616,630,732,750,756,768,810,828,910,936,982,1032,1060,1128,1216,
%U 1302,1366,1558,1626,1696,1698,1758,1800,1810,1830,1932,1996,2002,2026,2080
%N Positive integers k such that k^7 + 1 is semiprime.
%C 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.
%H Robert Price, <a href="/A105041/b105041.txt">Table of n, a(n) for n = 1..1414</a>
%F 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.
%e n n^7+1 = ((n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
%e 2 129 = 3 x 43
%e 10 10000001 = 11 * 909091
%e 16 268435457 = 17 * 15790321
%e 18 612220033 = 19 * 32222107
%e 46 435817657217 = 47 * 9272716111
%t Select[Range[0,200000], PrimeQ[# + 1] && PrimeQ[(#^7 + 1)/(# + 1)] &] (* _Robert Price_, Mar 11 2015 *)
%t Select[Range[2500], Plus@@Last/@FactorInteger[#^7 + 1]==2 &] (* _Vincenzo Librandi_, Mar 12 2015 *)
%t Select[Range[2100],PrimeOmega[#^7+1]==2&] (* _Harvey P. Dale_, Jun 18 2019 *)
%o (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
%o (PARI) is(n)=isprime(n+1) && isprime((n^7+1)/(n+1)) \\ _Charles R Greathouse IV_, Aug 31 2021
%Y Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Apr 03 2005
%E More terms from _R. J. Mathar_, Dec 14 2009
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