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A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments. 3
13, 11, 19, 17, 31, 29, 43, 41, 67, 79, 103, 101, 113, 139, 137, 173, 223, 257, 283, 281, 317, 353, 367, 401, 439, 499, 607, 619, 617, 643, 641, 653, 677, 761, 787, 823, 821, 907, 941, 977, 1039, 1087, 1181, 1193, 1361, 1373, 1399, 1433, 1447, 1543, 1579 (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..51.

EXAMPLE

The POBAs to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

MATHEMATICA

x = 6; 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, A265770/A265771 *)

Numerator[tL]   (* A227756 *)

Denominator[tL] (* A158015 *)

Numerator[tU]   (* A051644 *)

Denominator[tU] (* A007693 *)

Numerator[y]    (* A222570 *)

Denominator[y]  (* A265771 *)

CROSSREFS

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265771.

Sequence in context: A158956 A259663 A160130 * A206607 A196124 A196121

Adjacent sequences:  A265767 A265768 A265769 * A265771 A265772 A265773

KEYWORD

nonn,frac

AUTHOR

Clark Kimberling, Dec 20 2015

STATUS

approved

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Last modified February 27 19:19 EST 2020. Contains 332308 sequences. (Running on oeis4.)