A224487 Least integer b > F(n) such that sum_{k=1}^n F(k)*b^{k-1} is prime, where F = A000045.

2, 4, 4, 6, 10, 39, 102, 44, 165, 96, 154, 446, 406, 714, 999, 1634, 2698, 5445, 7630, 11670, 17833, 28758, 46686, 75178, 121782, 197890, 319081, 522734, 840924

Offset: 2

Author

Zhi-Wei Sun, Apr 08 2013

Keywords

• nonn

Comments

Conjecture: For any n > 1 we have a(n) < F(n+4); moreover, there are infinitely many integers b > F(n) such that sum_{k=1}^n F(k)*b^{k-1} is prime.

Examples

			a(6) = 10 since sum_{k=0}^6 F(k)*10^{k-1} = 853211 is prime but  sum_{k=0}^6 F(k)*9^{k-1} = 507556 is composite.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..49

Crossrefs

Cf. A000040, A000045, A220947, A217785, A218465, A217788.

Programs

Mathematica

A[n_,x_]:=A[n,x]=Sum[Fibonacci[k]*x^(k-1),{k,1,n}]
Do[Do[Do[If[PrimeQ[A[n,s]]==True,Print[n," ",s];Goto[aa]],{s,Fibonacci[n]+1,Fibonacci[n+4]-1}];
Print[n," ",counterexample];Label[aa];Continue,{n,2,20}]]