A051628 Number of digits in decimal expansion of 1/n before the periodic part begins.

0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 4, 0, 1, 0, 2, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 0, 5, 0, 1, 1, 2, 0, 1, 0, 3, 0, 1, 0, 2, 1, 1, 0, 4, 0, 2, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 0, 1, 0, 6, 1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 5, 0, 1, 0, 2, 0, 1, 0, 3, 1

Offset

1,4

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Luis Lopes, POLYA013: Length of the nonperiodic part of the decimal expansion of 1/N, The Problem Center, POLYA, 2001. [Wayback Machine link]

Index entries for sequences related to decimal expansion of 1/n.

Formula

For n>1, a(n) = max(i, j) where n=2^i*3^x*5^j*... is the prime decomposition of n.

From Amiram Eldar, Aug 25 2024: (Start)

a(n) = max(A007814(n), A112765(n)).

a(n) = A051903(A132741(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 41/36. (End)

Example

1/8 = .1250000... so a(8)=3, 1/15 = .0666666..., so a(15)=1.

Mathematica

a[n_] := Max[IntegerExponent[n, 2], IntegerExponent[n, 5]];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jul 20 2022, after Chai Wah Wu *)

Prog

(Python)
from sympy import multiplicity
def A051628(n): return max(multiplicity(2, n), multiplicity(5, n)) # Chai Wah Wu, Feb 07 2022
(PARI) a(n) = max(valuation(n, 2), valuation(n, 5)); \\ Michel Marcus, Oct 27 2022

Crossrefs

Cf. A007814, A051903, A112765, A132741.

Sequence in context: A110962 A065715 A180984 * A163540 A127967 A147602

Adjacent sequences: A051625 A051626 A051627 * A051629 A051630 A051631

Keyword

nonn,nice,easy,base

Author

N. J. A. Sloane

Extensions

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

More terms from Franklin T. Adams-Watters, May 05 2006

Status

approved