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A002523 a(n) = n^4 + 1. 29
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 (list; graph; refs; listen; history; text; internal format)
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

Mabkhout, M. (1993). "Minoration de P(x^4+1)". Rend. Sem. Fac. Sci. Univ. Cagliari 63 (2): 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 to values of cyclotomic polynomials of integer argument

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

FORMULA

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) - R. J. Mathar, Apr 28 2008

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

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

(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

CROSSREFS

Cf. A005117.

Sequence in context: A183175 A060352 A215185 * A079889 A053786 A181546

Adjacent sequences:  A002520 A002521 A002522 * A002524 A002525 A002526

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 20 02:29 EST 2018. Contains 297938 sequences.