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

8424432925592889329288197322308900672459420460792433, 17361015163508605989239159575667846308252873717727992, 26297597401424322649190121829026791944046326974663551, 35234179639340039309141084082385737579839780231599110

Offset: 1

Author

Ilan Vardi, Stan Wagon

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A118119, A255859.

Programs

Mathematica

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

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

Formula

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

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

Extensions

More terms from Max Alekseyev, Jul 26 2009