OFFSET
0,2
COMMENTS
a(n) = Phi_8(n), where Phi_k is the k-th cyclotomic polynomial.
All odd prime factors of a(n) are congruent to 1 modulo 8. - Nick Hobson, Jan 14 2007
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
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
REFERENCES
M. Mabkhout, "Minoration de P(x^4+1)", Rend. Sem. Fac. Sci. Univ. Cagliari 63 (2) (1993), 135-148.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jung-Jo Lee and M. Ram Murty, Dirichlet series and hyperelliptic curves, Forum Math. 19 (2007), 677-705.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
From R. J. Mathar, Apr 28 2008: (Start)
O.g.f.: (1 - 3*x + 17*x^2 + 7*x^3 + 2*x^4)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
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
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
Product_{n>=1} (1 - 1/a(n)) = 2*Pi^2/(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)). - Amiram Eldar, Jan 26 2024
MAPLE
A002523 := proc(n)
numtheory[cyclotomic](8, n) ;
end proc:
seq(A002523(n), n=0..20) ; # R. J. Mathar, Feb 07 2014
MATHEMATICA
Table[n^4+1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 2, 17, 82, 257}, 30] (* Ray Chandler, Aug 26 2015 *)
PROG
(Magma) [n^4 + 1: n in [0..40]]; // Vincenzo Librandi, Jun 07 2011
(PARI) a(n)=n^4+1 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved