A265760 - OEIS

%I #16 Dec 20 2015 13:50:10

%S 2,3,7,11,13,17,19,29,31,41,43,59,61,71,73,101,103,107,109,137,139,

%T 149,151,179,181,191,193,197,199,227,229,239,241,269,271,281,283,311,

%U 313,347,349,419,421,431,433,461,463,521,523,569,571,599,601,617,619

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

%C 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, ...).

%C See A265772 and A265774 for definitions of lower POBA and upper POBA, and also a guide to selected POBA sequences.

%C With the exception of the 2 and the 5 this seems to be the same as A001097. - _R. J. Mathar_, Dec 19 2015

%e 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.

%t x = 1; z = 200; p[k_] := p[k] = Prime[k];

%t t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];

%t d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)

%t t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];

%t d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)

%t v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];

%t b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];

%t y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)

%t Numerator[tL]   (* A001359 *)

%t Denominator[tL] (* A006512 *)

%t Numerator[tU]   (* A006512 *)

%t Denominator[tU] (* A001359 *)

%t Numerator[y]    (* A265759 *)

%t Denominator[y]  (* A265760 *)

%Y Cf. A000040, A001359, A006512, A265759.

%K nonn,frac

%O 1,1

%A _Clark Kimberling_, Dec 15 2015