A002523 a(n) = n^4 + 1.

1, 2, 17, 82, 257, 626, 1297, 2402, 4097, 6562, 10001, 14642, 20737, 28562, 38417, 50626, 65537, 83522, 104977, 130322, 160001, 194482, 234257, 279842, 331777, 390626, 456977, 531442, 614657, 707282, 810001, 923522, 1048577, 1185922, 1336337, 1500626, 1679617

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).
• Index to values of cyclotomic polynomials of integer argument.

Crossrefs

Cf. A000404, A005117, A111925.

Programs

Magma

[n^4 + 1: n in [0..40]]; // Vincenzo Librandi, Jun 07 2011

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 *)

Maxima

A002523(n):=n^4+1$ makelist(A002523(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI

a(n)=n^4+1 \\ Charles R Greathouse IV, Sep 24 2015

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