A010034 Numbers k such that gcd(k^17 + 9, (k+1)^17 + 9) > 1.

8424432925592889329288197322308900672459420460792433, 17361015163508605989239159575667846308252873717727992, 26297597401424322649190121829026791944046326974663551, 35234179639340039309141084082385737579839780231599110

Offset

1,1

Comments

In other words, let f(n) = gcd(n^17 + 9, (n+1)^17 + 9). Then f(n) = 1 for all n <= 8424432925592889329288197322308900672459420460792432, but f(8424432925592889329288197322308900672459420460792433) > 1.

In fact f(8424432925592889329288197322308900672459420460792433) = 8936582237915716659950962253358945635793453256935559.

Links

M. F. Hasler, Table of n, a(n) for n = 1..100

Tanya Khovanova, Recursive Sequences

Stan Wagon, Macalester College Problem of the week # 805, MacPOW archive on MathForum.org. Spring 1996.

Péter E. Frenkel, József Pelikán, On the greatest common divisor of the value of two polynomials, Amer. Math. Monthly 124:5 (2017), 446-450. arXiv:1608.07936 [math.NT]

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 8424432925592889329288197322308900672459420460792433 + 8936582237915716659950962253358945635793453256935559*(n-1). - Max Alekseyev, Jul 26 2009

a(1) = A255859(17). - M. F. Hasler, Mar 17 2015

Mathematica

Table[8424432925592889329288197322308900672459420460792433+ 8936582237915716659950962253358945635793453256935559(n-1), {n, 5}] (* or *) LinearRecurrence[{2, -1}, {8424432925592889329288197322308900672459420460792433, 17361015163508605989239159575667846308252873717727992}, 5] (* Harvey P. Dale, Jun 12 2014 *)

Prog

(PARI) A010034(n)=8936582237915716659950962253358945635793453256935559*n-512149312322827330662764931050044963334032796143126 \\ M. F. Hasler, Mar 17 2015
(PARI) \\ The values (a(1), p) can also be found using:
{p=polresultant(x^17+9, (x+1)^17+9); s=vector(2, i, Mod(-9, p)^(1/17)); (u=s[2]/s[1])!=1&&until(setsearch(Set(s=concat(s, s[#s]*u)), s[#s]+1), )}
\\ Then the last element s[#s] equals Mod(a(1), p). - M. F. Hasler, Mar 26 2015

Crossrefs

Cf. A118119, A255859.

Sequence in context: A095494 A246290 A095496 * A118329 A095498 A358419

Adjacent sequences: A010031 A010032 A010033 * A010035 A010036 A010037

Keyword

nonn,easy

Author

Ilan Vardi, Stan Wagon

Extensions

More terms from Max Alekseyev, Jul 26 2009

Status

approved