A304879 a(n) = denominator(min{r with r in R}) and R = {0 <= r rational <= 1 and [r*k] = n}. Here k = 10^(floor(log_10(n))+1) and [x] = floor(x+1/2) + ceiling((2*x-1)/4) - floor((2*x-1)/4) - 1.

1, 10, 6, 7, 8, 2, 9, 3, 4, 7, 21, 19, 26, 95, 37, 62, 58, 6, 40, 27, 41, 39, 93, 71, 17, 53, 98, 83, 40, 7, 61, 59, 73, 83, 98, 84, 76, 63, 8, 96, 81, 79, 53, 87, 85, 92, 90, 43, 40, 68, 2, 99, 33, 99, 71, 11, 9, 23, 40, 94, 42, 38, 13, 99, 74, 31, 29, 3, 40

Offset

0,2

Comments

a(n) is the denominator of the smallest nonnegative fraction r such that round(10^d*r) = n, where d is the number of digits of n and Gaussian rounding (round half to even) is applied.

Links

Table of n, a(n) for n=0..68.

Example

The table below shows the different rational numbers which satisfy the requirements of the definition except minimality. The denominators of the first rational number in each row constitute the sequence. Note that the round function is not implemented uniformly in popular software. For example, Mathematica  matches our definition, while Maple's round function would return incorrect values.
.
  n | rational numbers       decimal value   rounded(10*r)
----+---------------------------------------------------
  0 | 0/1,                   .0000000000,         0
  1 | 1/10, 1/9, 1/8, 1/7,   .1000000000,         1
  2 | 1/6, 1/5, 2/9, 1/4,    .1666666667,         2
  3 | 2/7, 3/10, 1/3,        .2857142857,         3
  4 | 3/8, 2/5, 3/7, 4/9,    .3750000000,         4
  5 | 1/2,                   .5000000000,         5
  6 | 5/9, 4/7, 3/5, 5/8,    .5555555556,         6
  7 | 2/3, 7/10, 5/7,        .6666666667,         7
  8 | 3/4, 7/9, 4/5, 5/6,    .7500000000,         8
  9 | 6/7, 7/8, 8/9, 9/10,   .8571428571,         9

Maple

r := proc(n) local nint, k, p, q, S; k := 10^(ilog10(n)+1);
nint := m -> floor(m + 1/2) +  ceil((2*m-1)/4) - floor((2*m-1)/4) - 1;
if n = 0 then return 0/1 fi; S := NULL;
for p from 1 to k do for q from p+1 to k do
    if nint(p*k/q) = n then S := S, p/q fi
od od; sort(convert({S}, list))[1] end:
a := n -> denom(r(n)): seq(a(n), n=0..68);

Crossrefs

Cf. A304880 (numerators), A303612.

Sequence in context: A158508 A102690 A076366 * A105155 A072930 A071358

Adjacent sequences: A304876 A304877 A304878 * A304880 A304881 A304882

Keyword

nonn,frac

Author

Peter Luschny, May 20 2018

Status

approved