

A265760


Denominators of primesonly best approximates (POBAs) to 1; see Comments.


2



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 primesonly 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



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]] &];


CROSSREFS



KEYWORD

nonn,frac


AUTHOR



STATUS

approved



