A265817 Denominators of upper primes-only best approximates (POBAs) to e; see Comments.

2, 5, 7, 11, 17, 29, 71, 4139, 5573, 6361, 9293, 17159, 18089, 2246039, 3135403, 3245939, 15812647, 23302423, 35724419, 36032933, 52372163, 107537039, 133106593, 167870293, 249402641, 260192623, 427246909, 475992263, 736166797, 1184975581, 1528278299, 2683676647, 5253849959, 5389332217

Offset

1,1

Comments

Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.

Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).

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

Links

Table of n, a(n) for n=1..34.

Example

The upper POBAs to e start with 77/2, 17/5, 23/7, 31/11, 47/17, 79/29, 193/71, 11251/4139.  For example, if p and q are primes and q > 71, and p/q > e, then 193/71 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, A265815, A265816, A265818, A265819.

Sequence in context: A038973 A197297 A172527 * A045349 A338339 A224320

Adjacent sequences: A265814 A265815 A265816 * A265818 A265819 A265820

Keyword

nonn,frac

Author

Clark Kimberling, Jan 06 2016

Extensions

More terms from Bert Dobbelaere, Jul 21 2022

Status

approved