A307070 a(n) is the number of decimal places before the decimal expansion of 1/n terminates, or the period of the recurring portion of 1/n if it is recurring.

0, 1, 1, 2, 1, 1, 6, 3, 1, 1, 2, 1, 6, 6, 1, 4, 16, 1, 18, 2, 6, 2, 22, 1, 2, 6, 3, 6, 28, 1, 15, 5, 2, 16, 6, 1, 3, 18, 6, 3, 5, 6, 21, 2, 1, 22, 46, 1, 42, 2, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 6, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13

Offset

1,4

Comments

If the decimal expansion of 1/n terminates, we will write it as ending with infinitely many 0's (rather than 9's). Then for any n > 1, the expansion of 1/n consists of a preamble whose length is given by A051628(n), followed by a periodic part with period length A007732(n). This sequence is defined as follows: If the only primes dividing n are 2 and 5 (see A003592), a(n) = A051628(n), otherwise a(n) = A007732(n) (and the preamble is ignored). - N. J. A. Sloane, Mar 22 2019

This sequence was discovered by a school class (aged 12-13) at Arden School, Solihull, UK.

Equally space the digits 0-9 on a circle. The digits of the decimal expansion of rational numbers can be connected on this circle to form data visualizations. This sequence is useful, cf. A007732 or A051626, for identifying the complexity of that visualization.

Links

Table of n, a(n) for n=1..79.

Example

1/1 is 1.0. There are no decimal digits, so a(1) = 0.
1/2 is 0.5. This is a terminating decimal. There is 1 digit, so a(2) = 1.
1/6 is 0.166666... This is a recurring decimal with a period of 1 (the initial '1' does not recur) so a(6) = 1.
1/7 is 0.142857142857... This is a recurring decimal, with a period of 6 ('142857') so a(7) = 6.

Prog

(Python)
def sequence(n):
..count = 0
..dividend = 1
..remainder = dividend % n
..remainders = [remainder]
..no_recurrence = True
..while remainder != 0:
....count += 1
....dividend = remainder * 10
....remainder = dividend % n
....if remainder in remainders:
......if no_recurrence:
........no_recurrence = False
........remainders = [remainder]
......else:
........return len(remainders)
....else:
......remainders.append(remainder)
..else:
....return count
(PARI) a(n) = my (t=valuation(n, 2), f=valuation(n, 5), r=n/(2^t*5^f)); if (r==1, max(t, f), znorder(Mod(10, r))) \\ Rémy Sigrist, May 08 2019

Crossrefs

See A114205 and A051628 for the preamble, A036275 and A051626 for the periodic part.

Cf. A001913, A003592, A054710, A121341, A122060.

Sequence in context: A372973 A156984 A181621 * A084268 A332405 A332403

Adjacent sequences: A307067 A307068 A307069 * A307071 A307072 A307073

Keyword

nonn,base,easy

Author

Luke W. Richards, Mar 22 2019

Extensions

More terms from Rémy Sigrist, May 08 2019

Status

approved