A265767 - OEIS

A265767

Numerators of upper primes-only best approximates (POBAs) to 5; see Comments.

11, 37, 67, 97, 157, 307, 337, 367, 397, 487, 547, 757, 787, 907, 967, 997, 1117, 1567, 1657, 1747, 1867, 1987, 2287, 2437, 2617, 2707, 2857, 2887, 3037, 3067, 3217, 3307, 3457, 3547, 3637, 3697, 3847, 4057, 4297, 4597, 4957, 4987, 5107, 5167, 5197, 5347

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OFFSET

1,1

COMMENTS

Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.

Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).

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

LINKS

Table of n, a(n) for n=1..46.

EXAMPLE

The upper POBAs to 5 start with 11/2, 37/7, 67/13, 97/19, 157/31, 307/61, 337/67, 367/73. For example, if p and q are primes and q > 19, and p/q > 5, then 97/19 is closer to 5 than p/q is.

MATHEMATICA

x = 5; 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, A265768/A265769 *)

Numerator[tL] (* A265766 *)

Denominator[tL] (* A158318 *)

Numerator[tU] (* A265767 *)