A343634 Number of the fraction which has (the digits of) n as repeating period in its decimal expansion, according to the canonical enumeration A038566/A038567.

23, 24, 3, 25, 26, 4, 27, 28, 1, 2951, 23, 327, 2952, 2953, 328, 2954, 2955, 34, 2956, 2957, 329, 24, 2958, 330, 2959, 2960, 35, 2961, 2962, 331, 2963, 2964, 3, 2965, 2966, 36, 2967, 2968, 332, 2969, 2970, 333, 2971, 25, 37, 2972, 2973, 334, 2974, 2975, 335, 2976, 2977, 38, 26

Offset

1,1

Comments

The fraction f = n/(10^L-1), where L is the number of decimal digits of n, has the infinite decimal expansion 0.{n}{n}{n}... (with special cases n = 9, 99 etc., where f = 0.999... = 1). This sequence lists the index of this fraction, for given n, corresponding to the "canonical enumeration of positive fractions <= 1", i.e., m such that f = A038566(m) / A038567(m-1).

Inspired by Angelini's blog post, which however takes a different approach to encoding fractions (in a more "decimal" way) and to deal with n's made of digits 9 or with repetitions, such as 11, 111, or 1010, etc.

Links

Table of n, a(n) for n=1..55.

Eric Angelini, Cantorisation, personal blog CinquanteSignes.blogspot.com, Oct 17 2021

Formula

a(n) = m such that A038566(m)/A038567(m-1) = n/A002283(A055642(n)), where A002283(1, 2, 3, ...) = (9, 99, 999, ...) and A055642(n) = number of digits of n.

Example

a(n = 1) = 23 because A038566(23)/A038567(22) = 1/9, the unique fraction (in lowest terms) whose decimal expansion is 0.111..., i.e., period = (1), repeated.
a(n = 2) = 24 because A038566(24)/A038567(23) = 2/9, the unique fraction (in lowest terms) whose decimal expansion is 0.222..., i.e., period = (2), repeated.
a(n = 3) = 3 because A038566(3)/A038567(2) = 1/3, the unique fraction (in lowest terms) whose decimal expansion is 0.333..., i.e., period = (3), repeated.
a(n = 9) = 1 because A038566(1)/A038567(0) = 1/1, the unique fraction (in lowest terms) equal to 0.999..., i.e., period = (9), repeated.
a(n = 10) = 2951 because A038566(2951)/A038567(2950) = 10/99, the unique fraction (in lowest terms) whose decimal expansion is 0.1010..., i.e., period = (10), repeated.
a(11) = a(1) because the unique fraction that has decimal expansion 0.1111..., i.e., period (11) repeated, is 1/9, the same as for 0.111..., i.e., period (1), repeated.

Prog

(PARI) apply( {A343634(n, d=10^(logint(n, 10)+1)-1, g=gcd(n, d)) = A002088(d\g-1) - sum(k=1, n\=g, gcd(k, d) > 1) + n}, [1..55])

Crossrefs

Cf. A038566/A038567, A002283, A055642, A002088.

Sequence in context: A040507 A022357 A165844 * A291511 A062999 A241889

Adjacent sequences: A343631 A343632 A343633 * A343635 A343636 A343637

Keyword

nonn,base

Author

M. F. Hasler, Oct 18 2021

Status

approved