A111002 - OEIS

%I #26 Sep 08 2022 08:45:20

%S 1,3,7,91,21,31,43,57,73,91,777,133,157,183,211,241,273,2149,343,381,

%T 421,463,507,553,4207,651,703,757,813,871,931,6951,1057,1123,1191,

%U 1261,1333,1407,10381,1561,1641,1723,1807,1893,1981,14497,2163,2257,2353

%N a(n) = gcd(f(n), f(n+1)) where f(n) = n^4 + n^2 + 1.

%H Harvey P. Dale, <a href="/A111002/b111002.txt">Table of n, a(n) for n = 0..1000</a>

%H PlanetMath, <a href="http://planetmath.org/ExampleOfGcd">Example of GCD</a>

%F a(n) = gcd(f(n), f(n+1)) for all n. a(n) = n^2 + n + 1, except when n congruent to 3 modulo 7 when a(n) = 7(n^2 + n + 1).

%F Conjectures from _Colin Barker_, Oct 06 2015: (Start)

%F a(n) = 3*a(n-7) - 3*a(n-14) + a(n-21) for n>20.

%F G.f.: -(x^20 +3*x^19 +7*x^18 +91*x^17 +21*x^16 +31*x^15 +43*x^14 +54*x^13 +64*x^12 +70*x^11 +504*x^10 +70*x^9 +64*x^8 +54*x^7 +43*x^6 +31*x^5 +21*x^4 +91*x^3 +7*x^2 +3*x +1) / ((x -1)^3*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)^3).

%F (End)

%e a(10) = 7(10^2 + 10 + 1) = 777 because 10 is congruent to 3 modulo 7.

%t f[n_] := n^4 + n^2 + 1; Table[ GCD[f[n], f[n + 1]], {n, 0, 49}] (* _Robert G. Wilson v_, Oct 02 2005 *)

%t GCD[#[[1]],#[[2]]]&/@Partition[Table[n^4+n^2+1,{n,0,50}],2,1] (* _Harvey P. Dale_, Mar 07 2015 *)

%o (PARI) m=50;a=3;for(k=2,m,b=k^4+k^2+1;print1(gcd(a,b),",");a=b) \\ _Klaus Brockhaus_, Oct 02 2005

%o (Magma) [Gcd(n^4+n^2+1, n^4+4*n^3+7*n^2+6*n+3): n in [0..50]]; // _Vincenzo Librandi_, Oct 07 2015

%K easy,nonn

%O 0,2

%A _Pahikkala Jussi_, Sep 30 2005

%E More terms from _Robert G. Wilson v_ and _Klaus Brockhaus_, Oct 02 2005