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 A265815 Denominators of lower primes-only best approximates (POBAs) to e; see Comments. 7
 2, 5, 7, 113, 163, 227, 823, 887, 1093, 2179, 2591, 2797, 4373, 4657, 5651, 8867, 27673, 32749, 47189, 104459, 155723, 430061, 583853, 673297, 1126523, 1869173, 3120317, 3445919, 8341961, 24681191, 26349383, 70271051, 77869361, 81514259, 89910487, 157461181, 533931763, 583892083, 770930497 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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. For a guide to POBAs, lower POBAs, and upper POBAs, see A265759. LINKS Table of n, a(n) for n=1..39. EXAMPLE 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. 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 *) CROSSREFS Cf. A000040, A265759, A265814, A265816, A265817, A265818, A265819. Sequence in context: A343833 A103056 A041445 * A041961 A242169 A058854 Adjacent sequences: A265812 A265813 A265814 * A265816 A265817 A265818 KEYWORD nonn,frac AUTHOR Clark Kimberling, Jan 02 2016 EXTENSIONS More terms from Bert Dobbelaere, Jul 21 2022 STATUS approved

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Last modified June 16 13:05 EDT 2024. Contains 373429 sequences. (Running on oeis4.)