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A265819
Denominators of primes-only best approximates (POBAs) to e; see Comments.
7
2, 2, 5, 7, 71, 2591, 2797, 4139, 5573, 5651, 6361, 8867, 17159, 18089, 1126523, 1869173, 3120317, 3245939, 8341961, 23302423, 24681191, 26349383, 52372163, 81514259, 89910487, 157461181, 475992263, 583892083, 770930497, 1184975581, 1436753887, 1498302107, 5389332217
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 POBA's. 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 A265759 for a guide to related sequences. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).
EXAMPLE
The POBAs to Pi start with 7/2, 5/2, 13/5, 19/7, 193/71, 7043/2591, 7603/2797. For example, if p and q are primes and q > 2591, then 7043/2591 is closer to e than p/q is.
MATHEMATICA
x = E; z = 1000; 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, A265818/A265819 *)
Numerator[tL] (* A265814 *)
Denominator[tL] (* A265815 *)
Numerator[tU] (* A265816 *)
Denominator[tU] (* A265817 *)
Numerator[y] (* A265818 *)
Denominator[y] (* A265819 *)
KEYWORD
nonn,frac,more
AUTHOR
Clark Kimberling, Jan 06 2016
EXTENSIONS
More terms from Bert Dobbelaere, Jul 21 2022
STATUS
approved