A265796 Numerators of lower primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

3, 11, 37, 163, 173, 241, 571, 1231, 1571, 2351, 3571, 25463, 69247

Offset: 1

Author

Clark Kimberling, Dec 29 2015

Keywords

• nonn
• frac
• more

Comments

Suppose that x > 0. A fraction p/q of primes is a lower primes-only best approximate, and we write "p/q is in L(x)", if u/v < p/q < x < p'/q for all primes u and v such that v < q, where p' is least prime > p.

Let q(1) be the least prime q such that u/q < x for some prime u, and let p(1) be the greatest such u. The sequence L(x) follows inductively: for n > 1, let q(n) is the least prime q such that p(n)/q(n) < p/q < x for some prime p. Let q(n+1) = q and let p(n+1) be the greatest prime p such that p(n)/q(n) < p/q < x.

For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.

Examples

			The lower POBAs to tau start with 3/2, 11/7, 37/23, 163/101, 173/107, 241/149. For example, if p and q are primes and q > 101, and p/q < tau, then 163/101 is closer to tau than p/q is.
		

Crossrefs

Cf. A000040, A001622, A265759, A265797, A265798, A265799, A265800, A265801.

Programs

Mathematica

x = GoldenRatio; 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, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

Extensions

a(12)-a(13) from Robert Price, Apr 06 2019