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A010034 Numbers k such that gcd(k^17 + 9, (k+1)^17 + 9) > 1. 2

%I

%S 8424432925592889329288197322308900672459420460792433,

%T 17361015163508605989239159575667846308252873717727992,

%U 26297597401424322649190121829026791944046326974663551

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

%C 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.

%C In fact f(8424432925592889329288197322308900672459420460792433) = 8936582237915716659950962253358945635793453256935559.

%H M. F. Hasler, <a href="/A010034/b010034.txt">Table of n, a(n) for n = 1..100</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H S. Wagon, <a href="http://mathforum.org/wagon/spring96/p805.html">Macalester College Problem of the week # 805</a>, MacPOW archive on MathForum.org. Spring 1996.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).

%F a(n) = 8424432925592889329288197322308900672459420460792433 + 8936582237915716659950962253358945635793453256935559*(n-1). - _Max Alekseyev_, Jul 26 2009

%F a(1) = A255859(17). - _M. F. Hasler_, Mar 17 2015

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

%o (PARI) A010034(n)=8936582237915716659950962253358945635793453256935559*n-512149312322827330662764931050044963334032796143126 \\ _M. F. Hasler_, Mar 17 2015

%o (PARI) \\ The values (a(1),p) can also be found using:

%o {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),)}

%o \\ Then the last element s[#s] equals Mod(a(1),p). - _M. F. Hasler_, Mar 26 2015

%Y Cf. A118119, A255859.

%K nonn,easy,bref

%O 1,1

%A Ilan Vardi, _Stan Wagon_

%E More terms from _Max Alekseyev_, Jul 26 2009

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Last modified May 11 13:41 EDT 2021. Contains 343791 sequences. (Running on oeis4.)