A265768 - OEIS

A265768

Numerators of primes-only best approximates (POBAs) to 5; see Comments.

7, 11, 23, 37, 53, 67, 83, 97, 113, 157, 233, 263, 293, 307, 337, 353, 367, 397, 443, 487, 503, 547, 563, 653, 683, 743, 757, 787, 863, 907, 953, 967, 983, 997, 1117, 1163, 1193, 1283, 1553, 1567, 1583, 1657, 1733, 1747, 1867, 1913, 1987, 2003, 2153, 2213

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OFFSET

1,1

COMMENTS

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..50.

EXAMPLE

The POBAs to 5 start with 7/2, 11/2, 23/5, 37/7, 53/11, 67/13, 83/17, 97/19, 113/23, 157/31, 233/47. For example, if p and q are primes and q > 13, then 67/13 is closer to 5 than p/q is.

MATHEMATICA

x = 5; z = 200; p[k_] := p[k] = Prime[k];

t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];

d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)

t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];

d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)

v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];

b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];

y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265768/A265769 *)

Numerator[tL] (* A265766 *)

Denominator[tL] (* A158318 *)

Numerator[tU] (* A265767 *)