cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

Original entry on oeis.org

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617
Offset: 1

Views

Author

Clark Kimberling, Dec 15 2015

Keywords

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. Note that for some choices of x, there are values of q for which there are two POBAs. 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 A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x Lower POBA Upper POBA POBA
1 A001359/A006512 A006512/A001359 A265759/A265760
3/2 A104163/A158708 A162336/A158709 A265761/A222565
2 A005383/A005382 A005385/A005384 A079149/A120628
3 A091180/A088878 A094525/A023208 A265763/A265764
4 A162857/A062737 A090866/A023212 A265765/A120639
5 A265766/A158318 A265767/A023217 A265768/A265769
6 A227756/A158015 A051644/A007693 A265770/A265771
sqrt(2) A265772/A265773 A265774/A265775 A265776/A265777
sqrt(3) A265778/A265779 A265780/A265781 A265782/A265783
sqrt(5) A265784/A265785 A265786/A265787 A265788/A265789
sqrt(8) A265790/A265791 A265792/A265793 A265794/A265795
tau A265796/A265797 A265798/A265799 A265800/A265801
1/tau A265799/A265798 A265797/A265796 A265806/A265807
pi A265808/A265809 A265810/A265811 A265812/A265813
e A265814/A265815 A265816/A265817 A265818/A265819

Examples

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Crossrefs

Cf. A000040, A001359, A006512, A265759, A265760.

Programs

  • Mathematica
    x = 1; z = 200; 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, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A265767 Numerators of upper primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

11, 37, 67, 97, 157, 307, 337, 367, 397, 487, 547, 757, 787, 907, 967, 997, 1117, 1567, 1657, 1747, 1867, 1987, 2287, 2437, 2617, 2707, 2857, 2887, 3037, 3067, 3217, 3307, 3457, 3547, 3637, 3697, 3847, 4057, 4297, 4597, 4957, 4987, 5107, 5167, 5197, 5347
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2015

Keywords

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.

Examples

			The upper POBAs to 5 start with 11/2, 37/7, 67/13, 97/19, 157/31, 307/61, 337/67, 367/73. For example, if p and q are primes and q > 19, and p/q > 5, then 97/19 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A023217, A158318, A265759, A265766, A222568, A265769.

Programs

  • Mathematica
    x = 5; z = 200; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)

A265766 Numerators of lower primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

7, 13, 23, 53, 83, 113, 233, 263, 293, 353, 443, 503, 563, 653, 683, 743, 863, 953, 983, 1163, 1193, 1283, 1553, 1583, 1733, 1913, 2003, 2153, 2213, 2243, 2333, 2393, 2543, 2843, 2963, 3083, 3203, 3413, 3593, 3803, 3863, 4133, 4283, 4643, 4703, 4733, 5153
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2015

Keywords

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.

Examples

			The lower POBAs to 5 start with 7/2, 13/3, 23/5, 53/11, 83/17, 113/23, 233/47. For example, if p and q are primes and q > 17, and p/q < 5, then 83/17 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A023217, A158318, A222568, A265759, A265767, A265769.

Programs

  • Mathematica
    x = 5; z = 200; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)

A265769 Denominators of primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

2, 2, 5, 7, 11, 13, 17, 19, 23, 31, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 109, 113, 131, 137, 149, 151, 157, 173, 181, 191, 193, 197, 199, 223, 233, 239, 257, 311, 313, 317, 331, 347, 349, 373, 383, 397, 401, 431, 443, 449, 457, 467, 479, 487, 509
Offset: 1

Views

Author

Clark Kimberling, Dec 20 2015

Keywords

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 POBAs. 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.

Examples

			The POBAs to 5 start with 7/2, 11/2, 23/5, 37/7, 53/11, 67/13, 83/17, 97/19, 113/23, 157/31, 233/47. For example, if p and q are primes and q > 13, then 67/13 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A265759, A265766, A158318, A265767, A023217, A265768.

Programs

  • Mathematica
    x = 5; z = 200; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)
Showing 1-4 of 4 results.

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