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A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments. 56
3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617 (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. 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 A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.

x        Lower POBA       Upper POBA       POBA

1        A001359/A006512  A006512/A001359  A265759/A265760

3/2      A104163/A158708  A162336/A158709  A265761/A222565

2        A005383/A005382  A005385/A005384  A079149/A120628

3        A091180/A088878  A094525/A023208  A265763/A265764

4        A162857/A062737  A090866/A023212  A265765/A120639

5        A265766/A158318  A265767/A023217  A265768/A265769

6        A227756/A158015  A051644/A007693  A265770/A265771

sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777

sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783

sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789

sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795

tau      A265796/A265797  A265798/A265799  A265800/A265801

1/tau    A265799/A265798  A265797/A265796  A265806/A265807

pi       A265808/A265809  A265810/A265811  A265812/A265813

e        A265814/A265815  A265816/A265817  A265818/A265819

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

MATHEMATICA

x = 1; 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, A265759/A265760 *)

Numerator[tL]   (* A001359 *)

Denominator[tL] (* A006512 *)

Numerator[tU]   (* A006512 *)

Denominator[tU] (* A001359 *)

Numerator[y]    (* A265759 *)

Denominator[y]  (* A265760 *)

CROSSREFS

Cf. A000040, A001359, A006512, A265759, A265760.

Sequence in context: A075146 A062941 A211018 * A057674 A092935 A137455

Adjacent sequences:  A265756 A265757 A265758 * A265760 A265761 A265762

KEYWORD

nonn,frac

AUTHOR

Clark Kimberling, Dec 15 2015

STATUS

approved

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Last modified June 28 09:49 EDT 2017. Contains 288813 sequences.