A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

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

Offset: 1

Author

Clark Kimberling, Dec 15 2015

Keywords

• nonn
• frac

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

Examples

			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.
		

Crossrefs

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

Programs

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 *)