A265759 -id:A265759 - cp's OEIS frontend

A265786 Numerators of upper primes-only best approximates (POBAs) to sqrt(5); see Comments.

5, 7, 43, 83, 293, 709, 937, 1259, 2131, 6791, 8951, 12721, 26683, 111667, 154841

Offset: 1

Clark Kimberling, Dec 23 2015

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.

			The upper POBAs to sqrt(5) start with 5/2, 7/3, 43/19, 83/37, 293/131, 709/317, 937/419. For example, if p and q are primes and q > 131, and p/q > sqrt(5), then 293/131 is closer to sqrt(5) than p/q is.

Cf. A000040, A265759, A265784, A265785, A265787, A265788, A265789.

x = Sqrt[5]; 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, A265782/A265783 *)
Numerator[tL]   (* A265784 *)
Denominator[tL] (* A265785 *)
Numerator[tU]   (* A265786 *)
Denominator[tU] (* A265787 *)
Numerator[y]    (* A222588 *)
Denominator[y]  (* A265789 *)

a(13)-a(15) from Robert Price, Apr 05 2019

A265787 Denominators of upper primes-only best approximates (POBAs) to sqrt(5); see Comments.

Original entry on oeis.org

2, 3, 19, 37, 131, 317, 419, 563, 953, 3037, 4003, 5689, 11933, 49939, 69247

Offset: 1

Clark Kimberling, Dec 26 2015

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.

			The upper POBAs to sqrt(5) start with 5/2, 7/3, 43/19, 83/37, 293/131, 709/317, 937/419. For example, if p and q are primes and q > 131, and p/q > sqrt(5), then 293/131 is closer to sqrt(5) than p/q is.

Cf. A000040, A265759, A265784, A265785, A265786, A265788, A265789.

x = Sqrt[5]; 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, A265782/A265783 *)
Numerator[tL]   (* A265784 *)
Denominator[tL] (* A265785 *)
Numerator[tU]   (* A265786 *)
Denominator[tU] (* A265787 *)
Numerator[y]    (* A265788 *)
Denominator[y]  (* A265789 *)

a(13)-a(15) from Robert Price, Apr 05 2019

A265788 Numerators of primes-only best approximates (POBAs) to sqrt(5); see Comments.

Original entry on oeis.org

3, 5, 7, 11, 29, 163, 199, 521, 3571, 26683, 111667, 150427, 154841

Offset: 1

Clark Kimberling, Dec 26 2015

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.

			The POBAs to sqrt(5) start with 3/2, 5/2, 7/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 89, then 199/89 is closer to sqrt(5) than p/q is.

Cf. A000040, A265759, A265784, A265785, A265786, A265788, A265789.

x = Sqrt[5]; 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, A265782/A265783 *)
Numerator[tL]   (* A265784 *)
Denominator[tL] (* A265785 *)
Numerator[tU]   (* A265786 *)
Denominator[tU] (* A265787 *)
Numerator[y]    (* A265788 *)
Denominator[y]  (* A265789 *)

a(10)-a(13) from Robert Price, Apr 05 2019

A265789 Denominators of primes-only best approximates (POBAs) to sqrt(5); see Comments.

Original entry on oeis.org

2, 2, 3, 5, 13, 73, 89, 233, 1597, 11933, 49939, 67273, 69247

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The POBAs to sqrt(5) start with 3/2, 5/2, 7/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 89, then 199/89 is closer to sqrt(5) than p/q is.

Cf. A000040, A265759, A265784, A265785, A265786, A265788, A265789.

x = Sqrt[5]; 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, A265782/A265783 *)
Numerator[tL]   (* A265784 *)
Denominator[tL] (* A265785 *)
Numerator[tU]   (* A265786 *)
Denominator[tU] (* A265787 *)
Numerator[y]    (* A265788 *)
Denominator[y]  (* A265789 *)

a(10)-a(13) from Robert Price, Apr 05 2019

A265790 Numerators of lower primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

5, 13, 19, 31, 223, 359, 461, 659, 6163, 8539, 8737, 25453, 32377, 35839

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The lower POBAs to sqrt(8) start with 5/2, 13/5, 19/7, 31/11, 223/79, 359/127. For example, if p and q are primes and q > 79, and p/q < sqrt(8), then 223/79 is closer to sqrt(8) than p/q is.

Cf. A000040, A265759, A265791, A265792, A265793, A265794, A265795.

x = Sqrt[8]; 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, A265782/A265783 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(12)-a(14) from Robert Price, Apr 06 2019

A265791 Denominators of lower primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

2, 5, 7, 11, 79, 127, 163, 233, 2179, 3019, 3089, 8999, 11447, 12671

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The lower POBAs to sqrt(8) start with 5/2, 13/5, 19/7, 31/11, 223/79, 359/127. For example, if p and q are primes and q > 79, and p/q < sqrt(8), then 223/79 is closer to sqrt(8) than p/q is.

Cf. A000040, A265759, A265790, A265792, A265793, A265794, A265795.

x = Sqrt[8]; 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, A265794/A265795 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(12)-a(14) from Robert Price, Apr 06 2019

A265792 Numerators of upper primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

7, 17, 23, 37, 167, 563, 727, 1123, 1321, 1847, 2803, 4517, 46027, 79657, 85229, 103099, 182657, 199373

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The upper POBAs to sqrt(8) start with 7/2, 17/5, 23/7, 37/13, 167/59, 563/199, 727/257, 1123/397. For example, if p and q are primes and q > 13, and p/q > sqrt(8), then 37/13 is closer to sqrt(8) than p/q is.

Cf. A000040, A265759, A265790, A265791, A265793, A265794, A265795.

x = Sqrt[8]; 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, A265794/A265795 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(13)-a(18) from Robert Price, Apr 06 2019

A265793 Denominators of upper primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

2, 5, 7, 13, 59, 199, 257, 397, 467, 653, 991, 1597, 16273, 28163, 30133, 36451, 64579, 70489

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The upper POBAs to sqrt(8) start with 7/2, 17/5, 23/7, 37/13, 167/59, 563/199, 727/257, 1123/397. For example, if p and q are primes and q > 13, and p/q > sqrt(8), then 37/13 is closer to sqrt(8) than p/q is.

Cf. A000040, A265759, A265790, A265791, A265792, A265794, A265795.

x = Sqrt[8]; 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, A265794/A265795 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(13)-a(18) from Robert Price, Apr 06 2019

A265794 Numerators of primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

7, 5, 13, 19, 31, 167, 359, 461, 659, 1847, 2803, 4517, 32377, 35839, 199373

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The POBAs to sqrt(8) start with 7/2, 5/2, 13/5, 19/7, 31/11, 167/59, 359/127, 461/163, 659/233. For example, if p and q are primes and q > 59, then 167/59 is closer to sqrt(8) than p/q is.

Cf. A000040, A265759, A265790, A265791, A265792, A265793, A265795.

x = Sqrt[8]; 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, A265794/A265795 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(13)-a(15) from Robert Price, Apr 06 2019

A265795 Denominators of primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

2, 2, 5, 7, 11, 59, 127, 163, 233, 653, 991, 1597, 11447, 12671, 70489

Offset: 1

Clark Kimberling, Dec 29 2015

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.

			The POBAs to sqrt(8) start with 7/2, 5/2, 13/5, 19/7, 31/11, 167/59, 359/127, 461/163, 659/233. For example, if p and q are primes and q > 59, then 167/59 is closer to sqrt(8) than p/q is.

Cf. A000040, A010466, A265759, A265790, A265791, A265792, A265793, A265794.

x = Sqrt[8]; 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, A265794/A265795 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(13)-a(15) from Robert Price, Apr 06 2019

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A265786 Numerators of upper primes-only best approximates (POBAs) to sqrt(5); see Comments.

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A265787 Denominators of upper primes-only best approximates (POBAs) to sqrt(5); see Comments.

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A265788 Numerators of primes-only best approximates (POBAs) to sqrt(5); see Comments.

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A265789 Denominators of primes-only best approximates (POBAs) to sqrt(5); see Comments.

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A265790 Numerators of lower primes-only best approximates (POBAs) to sqrt(8); see Comments.

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A265791 Denominators of lower primes-only best approximates (POBAs) to sqrt(8); see Comments.

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A265792 Numerators of upper primes-only best approximates (POBAs) to sqrt(8); see Comments.

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A265793 Denominators of upper primes-only best approximates (POBAs) to sqrt(8); see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica