A265772 - OEIS

%I #13 Apr 05 2019 17:45:07

%S 2,7,41,977,1093,1373,1721,2281,3121,3319,3947,4903,4937,8597,38287,

%T 64037,78643

%N Numerators of lower primes-only best approximates (POBAs) to sqrt(2); see Comments.

%C 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.

%C 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.

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

%e The lower POBAs to sqrt(2) start with 2/2, 7/5, 41/29, 977/691, 1093/773, 1373/971. For example, if p and q are primes and q > 691, and p/q < sqrt(2), then 977/691 is closer to sqrt(2) than p/q is.

%t x = Sqrt[2]; 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, A265776/A265777 *)

%t Numerator[tL]   (* A265772 *)

%t Denominator[tL] (* A265773 *)

%t Numerator[tU]   (* A265774 *)

%t Denominator[tU] (* A265775 *)

%t Numerator[y]    (* A265776 *)

%t Denominator[y]  (* A265777 *)

%Y Cf. A000040, A265759, A265773, A265774, A265775, A265776, A265777.

%K nonn,frac,more

%O 1,1

%A _Clark Kimberling_, Dec 20 2015

%E a(15)-a(17) from _Robert Price_, Apr 05 2019