This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002523 #77 Jan 26 2024 04:41:03
%S A002523 1,2,17,82,257,626,1297,2402,4097,6562,10001,14642,20737,28562,38417,
%T A002523 50626,65537,83522,104977,130322,160001,194482,234257,279842,331777,
%U A002523 390626,456977,531442,614657,707282,810001,923522,1048577,1185922,1336337,1500626,1679617
%N A002523 a(n) = n^4 + 1.
%C A002523 a(n) = Phi_8(n), where Phi_k is the k-th cyclotomic polynomial.
%C A002523 All odd prime factors of a(n) are congruent to 1 modulo 8. - _Nick Hobson_, Jan 14 2007
%C A002523 Lee and Murty, p. 685: "In spite of these remarkable advances, we are still unable to determine if n^4 + 1 is infinitely often a squarefree number". - _Jonathan Vos Post_, Sep 18 2007
%C A002523 Since a(n)*a(m) = (n^4+1)*(m^4+1) = ((n*m)^2-1)^2 + (n^2+m^2)^2, a(n)*a(m) is obvious member of A000404 for n*m > 1. Additionally, if m and n are the legs of a Pythagorean triple, then a(m)*a(n) is the member of A111925. - _Altug Alkan_, Apr 08 2016
%D A002523 M. Mabkhout, "Minoration de P(x^4+1)", Rend. Sem. Fac. Sci. Univ. Cagliari 63 (2) (1993), 135-148.
%H A002523 Vincenzo Librandi, <a href="/A002523/b002523.txt">Table of n, a(n) for n = 0..1000</a>
%H A002523 Jung-Jo Lee and M. Ram Murty, <a href="http://www.mast.queensu.ca/~murty/Lee-Murty-Forum.pdf">Dirichlet series and hyperelliptic curves</a>, Forum Math. 19 (2007), 677-705.
%H A002523 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%H A002523 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>.
%F A002523 From _R. J. Mathar_, Apr 28 2008: (Start)
%F A002523 O.g.f.: (1 - 3*x + 17*x^2 + 7*x^3 + 2*x^4)/(1-x)^5.
%F A002523 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%F A002523 Sum_{n>=0} 1/a(n) = 1/2 + Pi * (sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi)) / (2*sqrt(2) * (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))) = 1.578477579667136838318... . - _Vaclav Kotesovec_, Feb 14 2015
%F A002523 Sum_{n>=0} (-1)^n/a(n) = 1/2 - Pi * (cos(Pi/sqrt(2)) * sinh(Pi/sqrt(2)) + cosh(Pi/sqrt(2)) * sin(Pi/sqrt(2))) / (sqrt(2) * (cos(sqrt(2)*Pi) - cosh(sqrt(2)*Pi))) = 0.54942814871987317922929... . - _Vaclav Kotesovec_, Feb 14 2015
%F A002523 Product_{n>=1} (1 - 1/a(n)) = 2*Pi^2/(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)). - _Amiram Eldar_, Jan 26 2024
%p A002523 A002523 := proc(n)
%p A002523 numtheory[cyclotomic](8,n) ;
%p A002523 end proc:
%p A002523 seq(A002523(n),n=0..20) ; # _R. J. Mathar_, Feb 07 2014
%t A002523 Table[n^4+1,{n,0,60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)
%t A002523 LinearRecurrence[{5, -10, 10, -5, 1},{1, 2, 17, 82, 257},30] (* _Ray Chandler_, Aug 26 2015 *)
%o A002523 (Magma) [n^4 + 1: n in [0..40]]; // _Vincenzo Librandi_, Jun 07 2011
%o A002523 (Maxima) A002523(n):=n^4+1$ makelist(A002523(n),n,0,30); /* _Martin Ettl_, Nov 07 2012 */
%o A002523 (PARI) a(n)=n^4+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A002523 Cf. A000404, A005117, A111925.
%K A002523 nonn,easy
%O A002523 0,2
%A A002523 _N. J. A. Sloane_