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A265818 Numerators of primes-only best approximates (POBAs) to e; see Comments. 7
7, 5, 13, 19, 193, 7043, 7603, 11251, 15149, 15361, 17291, 24103, 46643, 49171 (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 POBA's. 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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).

LINKS

Table of n, a(n) for n=1..14.

EXAMPLE

The POBAs to Pi start with 7/2, 5/2, 13/5, 19/7, 193/71, 7043/2591, 7603/2797. For example, if p and q are primes and q > 2591, then 7043/2591 is closer to e than p/q is.

MATHEMATICA

x = E; z = 1000; 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, A265818/A265819 *)

Numerator[tL]   (* A265814 *)

Denominator[tL] (* A265815 *)

Numerator[tU]   (* A265816 *)

Denominator[tU] (* A265817 *)

Numerator[y]    (* A265818 *)

Denominator[y]  (* A265819 *)

CROSSREFS

Cf. A000040, A265759, A265814, A265815, A265816, A265817, A265819.

Sequence in context: A332768 A107323 A265794 * A046557 A298722 A299556

Adjacent sequences:  A265815 A265816 A265817 * A265819 A265820 A265821

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Jan 06 2016

STATUS

approved

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Last modified September 25 13:37 EDT 2021. Contains 347654 sequences. (Running on oeis4.)