cp's OEIS Frontend

A096172 Largest prime factor of n^4 + 1.

%I A096172 #37 Oct 28 2024 09:35:55
%S A096172 2,17,41,257,313,1297,1201,241,193,137,7321,233,14281,937,1489,65537,
%T A096172 41761,929,3833,160001,97241,3209,139921,331777,11489,26881,6481,
%U A096172 614657,353641,3361,1129,61681,6113,1336337,750313,98801,10529,50857,1156721
%N A096172 Largest prime factor of n^4 + 1.
%C A096172 Mabkhout shows that a(n) >= 137 for n > 3. - _Charles R Greathouse IV_, Apr 07 2014
%D A096172 Mustapha Mabkhout, Minoration de P(x^4+1), Rendiconti del Seminario della Facoltà di Scienze dell'Università di Cagliari 63:2 (1993), pp. 135-148.
%H A096172 Vincenzo Librandi and T. D. Noe, <a href="/A096172/b096172.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Vincenzo Librandi)
%H A096172 Florian Luca, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AAPASM_31_from19to24.pdf">Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1</a>, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
%H A096172 Marina Mureddu, <a href="http://www.numdam.org/item/AFST_1986-1987_5_8_2_109_0">A lower bound for P(x^4+1)</a>, Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 5, Vol. 8, No. 2 (1986-1987), pp. 109-119.
%F A096172 a(n) = A006530(1+n^4) = A014442(n^2). - _R. J. Mathar_, Jan 28 2017
%F A096172 From _Amiram Eldar_, Oct 28 2024: (Start)
%F A096172 a(n) > 113 for n > 3 (Mureddu, 1986-1987).
%F A096172 a(n) >= 233 for n >= 11 (Luca, 2004). (End)
%e A096172 a(1)=2 because 1^4 + 1 = 2;
%e A096172 a(2)=17: 2^4 + 1 = 17;
%e A096172 a(8)=241: 8^4 + 1 = 4097 = 17*241.
%t A096172 FactorInteger[#^4+1][[-1,1]]&/@Range[40] (* _Harvey P. Dale_, Apr 30 2012 *)
%o A096172 (PARI) a(n)=my(f=factor(n^4+1)[,1]); f[#f] \\ _Charles R Greathouse IV_, Apr 07 2014
%Y A096172 Cf. A000068, A002523, A014442, A037896, A006530, A069170, A096169, A096171.
%K A096172 nonn
%O A096172 1,1
%A A096172 _Hugo Pfoertner_, Jun 19 2004