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A265768 Numerators of primes-only best approximates (POBAs) to 5; see Comments. 5
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 (list; graph; refs; listen; history; text; internal format)
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 *)

Denominator[tU] (* A023217 *)

Numerator[y]    (* A222568 *)

Denominator[y]  (* A265769 *)

CROSSREFS

Cf. A000040, A023217, A158318, A265759, A265766, A265767, A265769.

Sequence in context: A175625 A082496 A239733 * A210001 A294074 A228227

Adjacent sequences:  A265765 A265766 A265767 * A265769 A265770 A265771

KEYWORD

nonn,frac

AUTHOR

Clark Kimberling, Dec 19 2015

STATUS

approved

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Last modified February 24 14:41 EST 2020. Contains 332209 sequences. (Running on oeis4.)