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A265816
Numerators of upper primes-only best approximates (POBAs) to e; see Comments.
7
7, 17, 23, 31, 47, 79, 193, 11251, 15149, 17291, 25261, 46643, 49171, 6105367, 8522909, 8823377, 42983231, 63342553, 97109039, 97947667, 142362299, 292315979, 361821233, 456318767, 677946667, 707276879, 1161377509, 1293881119, 2001108827, 3221097589, 4154291129, 7294989463, 14281444873
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.
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 *)
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Jan 02 2016
EXTENSIONS
More terms from Bert Dobbelaere, Jul 21 2022
STATUS
approved