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

2,1

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.

Links

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

Example

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.

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}]]

Crossrefs

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

Sequence in context: A351489 A022471 A359294 * A185342 A276985 A081238

Adjacent sequences: A224484 A224485 A224486 * A224488 A224489 A224490

Keyword

nonn

Author

Zhi-Wei Sun, Apr 08 2013

Status

approved