A265815 - OEIS

%I #10 Jul 21 2022 01:53:28

%S 2,5,7,113,163,227,823,887,1093,2179,2591,2797,4373,4657,5651,8867,

%T 27673,32749,47189,104459,155723,430061,583853,673297,1126523,1869173,

%U 3120317,3445919,8341961,24681191,26349383,70271051,77869361,81514259,89910487,157461181,533931763,583892083,770930497

%N Denominators of lower primes-only best approximates (POBAs) to e; 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 e; start with 5/2, 13/5, 19/7, 307/113, 443/163, 617/227, 2237/823. For example, if p and q are primes and q > 823, and p/q < e, then 2237/823 is closer to e than p/q is.

%t x = E; z = 1000; 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, A265818/A265819 *)

%t Numerator[tL]   (* A265814 *)

%t Denominator[tL] (* A265815 *)

%t Numerator[tU]   (* A265816 *)

%t Denominator[tU] (* A265817 *)

%t Numerator[y]    (* A265818 *)

%t Denominator[y]  (* A265819 *)

%Y Cf. A000040, A265759, A265814, A265816, A265817, A265818, A265819.

%K nonn,frac

%O 1,1

%A _Clark Kimberling_, Jan 02 2016

%E More terms from _Bert Dobbelaere_, Jul 21 2022