A307388 Length of the period of decimal representation of Product_{k=1..n} A038111(k)/A038110(k).

1, 27, 729, 59049, 43046721, 31381059609, 68630377364883, 150094635296999121, 328256967394537077627, 717897987691852588770249, 4710128697246244834921603689, 92709463147897837085761925410587, 3649600726280146254718103955713167842

Offset: 9

Author

Jamie Morken, Apr 06 2019

Keywords

• nonn
• base

Comments

The offset is 9 since for 0 < n < 5, the product is an integer, and for 4 < n < 9 the decimal expansion ends with zeros.

Examples

			For example for n=9 with (2/1) * (6/1) * (15/1) * (105/4) * (385/8) * (1001/16) * (17017/192) * (323323/3072) * (7436429/55296) = 2759414170256180364552625 / 154618822656 = 17846560482454.30745852273604315188195970323350694444444444444... so a(9) = 1.
		

Links

• Jamie Morken, Mathematica Stack Exchange question

Crossrefs

Cf. A038111, A038110, A051626, A058250.

Programs

Mathematica

Primorial[n_] := Times @@ Prime[Range[n]]
ClearAll[iter]
ClearAll[fracPer, vp];
(*p-adic order*)
vp[p_?PrimeQ, n_Integer] :=
  Length@NestWhileList[#/p &, n/p, IntegerQ] - 1;
(*fraction decimal expansion period*)
fracPer[q_Integer] := 0;
fracPer[q_Rational] := Module[{den, p2, p5}, den = Denominator[q];
   p2 = vp[2, den];
   p5 = vp[5, den];
   den = den/2^p2/5^p5;
   If[den == 1, 0, MultiplicativeOrder[10, den]]];
iter[{periods_, frac_, n_}] := {{periods, fracPer[#]}, #, n + 1} &[
   frac*Primorial[n]/EulerPhi[Primorial[Max[1, n - 1]]]];
Flatten@First@
  Nest[iter, {0, Primorial[0]/EulerPhi[Primorial[0]], 0}, 50]