A265759 -id:A265759 - cp's OEIS frontend

A265773 Denominators of lower primes-only best approximates (POBAs) to sqrt(2); see Comments.

2, 5, 29, 691, 773, 971, 1217, 1613, 2207, 2347, 2791, 3467, 3491, 6079, 27073, 45281, 55609

Offset: 1

Clark Kimberling, Dec 20 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(2) start with 2/2, 7/5, 41/29, 977/691, 1093/773, 1373/971. For example, if p and q are primes and q > 691, and p/q < sqrt(2), then 977/691 is closer to sqrt(2) than p/q is.

Cf. A000040, A265759, A265772, A265774, A265775, A265776, A265777.

x = Sqrt[2]; 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, A265776/A265777 *)
Numerator[tL]   (* A265772 *)
Denominator[tL] (* A265773 *)
Numerator[tU]   (* A265774 *)
Denominator[tU] (* A265775 *)
Numerator[y]    (* A265776 *)
Denominator[y]  (* A265777 *)

a(15)-a(17) from Robert Price, Apr 05 2019

A265775 Denominators of upper primes-only best approximates (POBAs) to sqrt(2); see Comments.

Original entry on oeis.org

2, 13, 37, 43, 139, 149, 313, 347, 593, 743, 883, 1009, 2617, 12269, 15731, 37879, 43789, 90533

Offset: 1

Clark Kimberling, Dec 20 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(2) start with 3/2, 19/13, 53/37, 61/43, 197/139, 211/149. For example, if p and q are primes and q > 139, and p/q > sqrt(2), then 197/139 is closer to sqrt(2) than p/q is.

Cf. A000040, A265759, A265772, A265773, A265774, A265776, A265777.

x = Sqrt[2]; 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, A265776/A265777 *)
Numerator[tL]   (* A265772 *)
Denominator[tL] (* A265773 *)
Numerator[tU]   (* A265774 *)
Denominator[tU] (* A265775 *)
Numerator[y]    (* A265776 *)
Denominator[y]  (* A265777 *)

a(14)-a(18) from Robert Price, Apr 05 2019

A265776 Numerators of primes-only best approximates (POBAs) to sqrt(2); see Comments.

Original entry on oeis.org

2, 3, 7, 41, 977, 1093, 1373, 1427, 3701, 8597, 22247, 38287, 53569, 61927, 78643

Offset: 1

Clark Kimberling, Dec 20 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(2) start with 2/2, 3/2, 7/5, 41/29, 977/691, 1093/773, 1373/971, 1427/1009. For example, if p and q are primes and q > 29, then 41/29 is closer to sqrt(2) than p/q is.

Cf. A000040, A265759, A265772, A265773, A265774, A265775, A265777.

x = Sqrt[2]; z = 800; 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, A265776/A265777 *)
Numerator[tL]   (* A265772 *)
Denominator[tL] (* A265773 *)
Numerator[tU]   (* A265774 *)
Denominator[tU] (* A265775 *)
Numerator[y]    (* A265776 *)
Denominator[y]  (* A265777 *)

a(11)-a(15) from Robert Price, Apr 05 2019

A265777 Denominators of primes-only best approximates (POBAs) to sqrt(2); see Comments.

Original entry on oeis.org

2, 2, 5, 29, 691, 773, 971, 1009, 2617, 6079, 15731, 27073, 37879, 43789, 55609

Offset: 1

Clark Kimberling, Dec 20 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(2) start with 2/2, 3/2, 7/5, 41/29, 977/691, 1093/773, 1373/971, 1427/1009. For example, if p and q are primes and q > 29, then 41/29 is closer to sqrt(2) than p/q is.

Cf. A000040, A265759, A265772, A265773, A265774, A265775, A265776.

x = Sqrt[2]; z = 800; 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, A265776/A265777 *)
Numerator[tL]   (* A265772 *)
Denominator[tL] (* A265773 *)
Numerator[tU]   (* A265774 *)
Denominator[tU] (* A265775 *)
Numerator[y]    (* A265776 *)
Denominator[y]  (* A265777 *)

a(11)-a(15) from Robert Price, Apr 05 2019

A265778 Numerators of lower primes-only best approximates (POBAs) to sqrt(3); see Comments.

Original entry on oeis.org

3, 5, 19, 71, 601, 1997, 2579, 3691, 75533, 167543, 175649

Offset: 1

Clark Kimberling, Dec 20 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(3) start with 3/2, 5/3, 19/11, 71/41, 601/347. For example, if p and q are primes and q > 347, and p/q < sqrt(3), then 601/347 is closer to sqrt(3) than p/q is.

Cf. A000040, A265759, A265779, A265780, A265781, A265782, A265783.

x = Sqrt[3]; 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]   (* A265778 *)
Denominator[tL] (* A265779 *)
Numerator[tU]   (* A265780 *)
Denominator[tU] (* A265781 *)
Numerator[y]    (* A262582 *)
Denominator[y]  (* A265783 *)

a(9)-a(11) from Robert Price, Apr 05 2019

A265779 Denominators of lower primes-only best approximates (POBAs) to sqrt(3); see Comments.

Original entry on oeis.org

2, 3, 11, 41, 347, 1153, 1489, 2131, 43609, 96731, 101411

Offset: 1

Clark Kimberling, Dec 20 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(3) start with 3/2, 5/3, 19/11, 71/41, 601/347. For example, if p and q are primes and q > 347, and p/q < sqrt(3), then 601/347 is closer to sqrt(3) than p/q is.

Cf. A000040, A265759, A265778, A265780, A265781, A265782, A265783.

x = Sqrt[3]; 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]   (* A265778 *)
Denominator[tL] (* A265779 *)
Numerator[tU]   (* A265780 *)
Denominator[tU] (* A265781 *)
Numerator[y]    (* A262582 *)
Denominator[y]  (* A265783 *)

a(9)-a(11) from Robert Price, Apr 05 2019

A265780 Numerators of upper primes-only best approximates (POBAs) to sqrt(3); see Comments.

Original entry on oeis.org

5, 7, 11, 13, 23, 83, 103, 127, 137, 227, 809, 1093, 1571, 4273, 5333, 16141, 20627, 41519, 56813, 111913

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.

Cf. A000040, A265759, A265778, A265779, A265781, A265782, A265783.

x = Sqrt[3]; 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]   (* A265778 *)
Denominator[tL] (* A265779 *)
Numerator[tU]   (* A265780 *)
Denominator[tU] (* A265781 *)
Numerator[y]    (* A262582 *)
Denominator[y]  (* A265783 *)

The upper POBAs to sqrt(3) start with 5/2, 7/3, 11/5, 13/7, 23/13, 83/47, 103/59. For example, if p and q are primes and q > 47, and p/q > sqrt(3), then 83/47 is closer to sqrt(3) than p/q is.

a(16)-a(20) from Robert Price, Apr 05 2019

A265781 Denominators of upper primes-only best approximates (POBAs) to sqrt(3); see Comments.

Original entry on oeis.org

2, 3, 5, 7, 13, 47, 59, 73, 79, 131, 467, 631, 907, 2467, 3079, 9319, 11909, 23971, 32801, 64613

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(3) start with 5/2, 7/3, 11/5, 13/7, 23/13, 83/47, 103/59. For example, if p and q are primes and q > 47, and p/q > sqrt(3), then 83/47 is closer to sqrt(3) than p/q is.

Cf. A000040, A265759, A265778, A265779, A265780, A265782, A265783.

x = Sqrt[3]; 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]   (* A265778 *)
Denominator[tL] (* A265779 *)
Numerator[tU]   (* A265780 *)
Denominator[tU] (* A265781 *)
Numerator[y]    (* A265782 *)
Denominator[y]  (* A265783 *)

a(16)-a(20) from Robert Price, Apr 05 2019

A265784 Numerators of lower primes-only best approximates (POBAs) to sqrt(5); see Comments.

Original entry on oeis.org

3, 5, 11, 29, 163, 199, 521, 3571, 91283, 150427

Offset: 1

Clark Kimberling, Dec 23 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(5) start with 3/2, 5/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 73, and p/q < sqrt(5), then 163/73 is closer to sqrt(5) than p/q is.

Cf. A000040, A265759, A265785, A265786, 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(9)-a(10) from Robert Price, Apr 05 2019

A265785 Denominators of lower primes-only best approximates (POBAs) to sqrt(5); see Comments.

Original entry on oeis.org

2, 3, 5, 13, 73, 89, 233, 1597, 40823, 67273

Offset: 1

Clark Kimberling, Dec 23 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(5) start with 3/2, 5/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 73, and p/q < sqrt(5), then 163/73 is closer to sqrt(5) than p/q is.

Cf. A000040, A265759, A265784, A265786, 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(9)-a(10) from Robert Price, Apr 05 2019

cp's OEIS Frontend

A265773 Denominators of lower primes-only best approximates (POBAs) to sqrt(2); see Comments.

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A265775 Denominators of upper primes-only best approximates (POBAs) to sqrt(2); see Comments.

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A265776 Numerators of primes-only best approximates (POBAs) to sqrt(2); see Comments.

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A265777 Denominators of primes-only best approximates (POBAs) to sqrt(2); see Comments.

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A265778 Numerators of lower primes-only best approximates (POBAs) to sqrt(3); see Comments.

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Mathematica

Extensions

A265779 Denominators of lower primes-only best approximates (POBAs) to sqrt(3); see Comments.

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Author

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Comments

Examples

Crossrefs

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Mathematica

Extensions

A265780 Numerators of upper primes-only best approximates (POBAs) to sqrt(3); see Comments.

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Mathematica

Formula

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A265781 Denominators of upper primes-only best approximates (POBAs) to sqrt(3); see Comments.

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Author

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Comments

Examples

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Programs

Mathematica