A303612 -id:A303612 - cp's OEIS frontend

A297367 Numerators of fraction whose denominator is defined in A303612.

0, 1, 1, 1, 2, 1, 3, 2, 3, 6, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 4, 1, 5, 3, 5, 2, 3, 4, 6, 1, 10, 6, 4, 7, 3, 7, 2, 7, 5, 3, 4, 5, 6, 7, 10, 17, 1, 18, 11, 8, 7, 6, 5, 4, 7, 10, 3, 11, 5, 12, 7, 11, 19, 2, 13, 9, 7, 5, 13, 8, 14, 3, 13, 10, 7, 11, 4

Offset: 0

Luca Petrone, Apr 30 2018

Cf. A303612.

# The function r is defined in A303612.
a := n -> numer(r(n)): seq(a(n), n=0..99); # Peter Luschny, May 19 2018

a = {1};
For[i = 1, i <= 100, i++,
nmax = 10^(Floor[Log[10, i]] + 1);
r = i/nmax;
For[n = 1, n <= nmax, n++,
If[Round[Round[n r]/n, 1/nmax] == r,
a = Flatten[Append[a, Round[n r]]];
Break[];
]]]

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

Original entry on oeis.org

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

Peter Luschny, May 20 2018

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.

			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

Cf. A304880 (numerators), A303612.

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);

cp's OEIS Frontend

A297367 Numerators of fraction whose denominator is defined in A303612.

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Mathematica

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

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Examples

Crossrefs

Programs

Maple

cp's OEIS Frontend

A297367 Numerators of fraction whose denominator is defined in A303612.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

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