A394354 a(n) = denominator q of the fraction p/q that minimizes |m/n - p/q| with m/n != p/q, where p is 1 or prime, q is prime, q < n, and m ranges from 1 to n - 1, with this rule to ensure uniqueness: if for given n there is more than one minimizing triple (m, p, q), then choose the greatest such m, then greatest q, then greatest p.

2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 11, 13, 17, 13, 19, 17, 19, 19, 23, 23, 19, 23, 19, 23, 29, 23, 31, 29, 29, 31, 23, 31, 37, 37, 31, 31, 41, 37, 43, 41, 43, 43, 47, 47, 47, 41, 47, 43, 53, 47, 41, 53, 47, 47, 59, 59, 61, 61, 61, 53, 59, 59, 67, 61, 59, 67, 71, 67, 73, 73, 59, 71, 73, 73, 79

Offset

3,1

Links

Table of n, a(n) for n=3..80.

Example

In the following table, columns 1-4 give values for the closest prime-fraction p/q to m/n as determined by the definition of a(n). In particular, a(3)-a(30) occupy column 2. Column 5 shows a comparison: < if p/q < m/n and > is p/q > m/n. Column 6 gives an approximation for the minimal distance |m/n - p/q|.
   n   m   p   q  comp.  min. dist.
   3   2   1   2    <    0.16666667
   4   3   2   3    <    0.08333333
   5   3   2   3    >    0.06666667
   6   1   1   5    >    0.03333333
   7   4   3   5    >    0.02857143
   8   1   1   7    >    0.01785714
   9   4   3   7    <    0.01587302
  10   7   5   7    >    0.01428571
  11   8   5   7    <    0.01298701
  12   1   1  11    >    0.00757576
  13   6   5  11    <    0.00699301
  14   1   1  13    >    0.00549451
  15   8   7  13    >    0.00512821
  16   3   2  11    <    0.00568182
  17   4   3  13    <    0.00452489
  18   1   1  17    >    0.00326797
  19  16  11  13    >    0.00404858
  20   1   1  19    >    0.00263158
  21  16  13  17    >    0.00280112
  22  15  13  19    >    0.00239234
  23   6   5  19    >    0.00228833
  24   1   1  23    >    0.00181159
  25  12  11  23    <    0.00173913
  26  15  11  19    >    0.00202429
  27  20  17  23    <    0.00161031
  28  25  17  19    >    0.00187970
  29  24  19  23    <    0.00149925
  30   1   1  29    >    0.00114943
|3/4 - 1/3| = 0.08333333... < |3/4 - p/q| for all other p/q as in the definition of a(n), so a(4) = 3.

Mathematica

u[p_] := p == 1 || PrimeQ[p];
t = Table[min = Infinity;
   b = {None, None, None};
   Do[Do[If[PrimeQ[q],
      Do[If[u[p], If[m/n != p/q, d = Abs[m/n - p/q];
         If[d < min, min = d;
          b = {m, p, q}; ]]], {p, n, 1, -1}]], {q, n - 1, 2, -1}], {m, n - 1, 1, -1}];
   {mVal, pVal, qVal} = b;
   comp = If[pVal/qVal < mVal/n, "<", ">"];
   {n, mVal, pVal, qVal, comp, NumberForm[N[min], {10, 8}]}, {n, 3, 30}];
TableForm[t, TableHeadings -> {None, {"n", "m", "p", "q", "comp.", "min. dist."}}]

Crossrefs

Cf. A000040, A007917, A385925, A385926, A385927, A394355, A394356.

Sequence in context: A093074 A284412 A385925 * A136548 A007917 A151799

Adjacent sequences: A394351 A394352 A394353 * A394355 A394356 A394357

Keyword

nonn,frac

Author

Clark Kimberling, Mar 17 2026

Status

approved