A265760 - OEIS

A265760

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

2, 3, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619

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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 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, and also a guide to selected POBA sequences.

With the exception of the 2 and the 5 this seems to be the same as A001097. - R. J. Mathar, Dec 19 2015

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 *)