A265759 -id:A265759 - cp's OEIS frontend

A265782 Numerators of primes-only best approximates (POBAs) to sqrt(3); see Comments.

5, 3, 5, 19, 71, 601, 1571, 2579, 3691, 56813, 111913

Offset: 1

Clark Kimberling, Dec 23 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(3) start with 5/2, 3/2, 5/3, 19/11, 71/41, 601/347, 1571/907. For example, if p and q are primes and q > 347, then 601/347 is closer to sqrt(3) than p/q is.

Cf. A000040, A265759, A265778, A265779, A265780, A265781, 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(10)-a(11) from Robert Price, Apr 05 2019

A265783 Denominators of primes-only best approximates (POBAs) to sqrt(3); see Comments.

Original entry on oeis.org

2, 2, 3, 11, 41, 347, 907, 1489, 2131, 32801, 64613

Offset: 1

Clark Kimberling, Dec 23 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(3) start with 5/2, 3/2, 5/3, 19/11, 71/41, 601/347, 1571/907. For example, if p and q are primes and q > 347, then 601/347 is closer to sqrt(3) than p/q is.

Cf. A000040, A265759, A265778, A265779, A265780, A265781, A265782.

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

A265801 Denominators of primes-only best approximates (POBAs) to the golden ratio, tau; see Comments.

Original entry on oeis.org

2, 2, 3, 7, 19, 23, 29, 97, 353, 563, 631, 919, 1453, 2207, 15271, 15737, 42797, 49939

Offset: 1

Clark Kimberling, Jan 02 2016

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.

Is this related to A165571? - R. J. Mathar, Jan 10 2016

			The POBAs to tau start with 5/2, 3/2, 5/3, 11/7, 31/19, 37/23, 47/29, 157/97, 571/353, 911/563. For example, if p and q are primes and q > 29, then 47/29 is closer to tau than p/q is.

Cf. A000040, A265759, A265796, A265797, A265798, A265799, A265800.

x = GoldenRatio; 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, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

a(15)-a(18) from Robert Price, Apr 06 2019

A265796 Numerators of lower primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

Original entry on oeis.org

3, 11, 37, 163, 173, 241, 571, 1231, 1571, 2351, 3571, 25463, 69247

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 tau start with 3/2, 11/7, 37/23, 163/101, 173/107, 241/149. For example, if p and q are primes and q > 101, and p/q < tau, then 163/101 is closer to tau than p/q is.

Cf. A000040, A001622, A265759, A265797, A265798, A265799, A265800, A265801.

x = GoldenRatio; 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, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

a(12)-a(13) from Robert Price, Apr 06 2019

A265797 Denominator of lower primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

Original entry on oeis.org

2, 7, 23, 101, 107, 149, 353, 761, 971, 1453, 2207, 15737, 42797

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 tau start with 3/2, 11/7, 37/23, 163/101, 173/107, 241/149. For example, if p and q are primes and q > 101, and p/q < tau, then 163/101 is closer to tau than p/q is.

Cf. A000040, A001622, A265759, A265796, A265798, A265799, A265800, A265801.

x = GoldenRatio; 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,A265800/A265801*)
Numerator[tL]   (*A265796*)
Denominator[tL] (*A265797*)
Numerator[tU]   (*A265798*)
Denominator[tU] (*A265799*)
Numerator[y]    (*A265800*)
Denominator[y]  (*A265801*)

a(12)-a(13) from Robert Price, Apr 06 2019

A265798 Numerators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

Original entry on oeis.org

5, 5, 31, 47, 157, 911, 1021, 1487, 4283, 5147, 8629, 11069, 15017, 20939, 22447, 24709, 38239, 80803

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 tau start with 5/2, 5/3, 31/19, 47/29, 157/97, 911/563, 1021/631. For example, if p and q are primes and q > 97, and p/q > tau, then 157/97 is closer to tau than p/q is.

Cf. A000040, A001622, A265759, A265796, A265797, A265799, A265800, A265801.

x = GoldenRatio; 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, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

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

A265799 Denominators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

Original entry on oeis.org

2, 3, 19, 29, 97, 563, 631, 919, 2647, 3181, 5333, 6841, 9281, 12941, 13873, 15271, 23633, 49939

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 tau start with 5/2, 5/3, 31/19, 47/29, 157/97, 911/563, 1021/631. For example, if p and q are primes and q > 97, and p/q > tau, then 157/97 is closer to tau than p/q is.

Cf. A000040, A001622, A265759, A265796, A265797, A265798, A265800, A265801.

x = GoldenRatio; 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, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

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

A265772 Numerators of lower primes-only best approximates (POBAs) to sqrt(2); see Comments.

Original entry on oeis.org

2, 7, 41, 977, 1093, 1373, 1721, 2281, 3121, 3319, 3947, 4903, 4937, 8597, 38287, 64037, 78643

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, A265773, 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

A265774 Numerators of upper primes-only best approximates (POBAs) to sqrt(2); see Comments.

Original entry on oeis.org

3, 19, 53, 61, 197, 211, 443, 491, 839, 1051, 1249, 1427, 3701, 17351, 22247, 53569, 61927, 128033

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

A265800 Numerators of primes-only best approximates (POBAs) to the golden ratio, tau; see Comments.

Original entry on oeis.org

5, 3, 5, 11, 31, 37, 47, 157, 571, 911, 1021, 1487, 2351, 3571, 24709, 25463, 69247, 80803

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.

How is this related to A165572? - R. J. Mathar, Jan 10 2016

			The POBAs to tau start with 5/2, 3/2, 5/3, 11/7, 31/19, 37/23, 47/29, 157/97, 571/353, 911/563. For example, if p and q are primes and q > 29, then 47/29 is closer to tau than p/q is.

Cf. A000040, A265759, A265796, A265797, A265798, A265799, A265801.

x = GoldenRatio; 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, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

a(15)-a(18) from Robert Price, Apr 06 2019

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A265782 Numerators of primes-only best approximates (POBAs) to sqrt(3); see Comments.

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A265783 Denominators of primes-only best approximates (POBAs) to sqrt(3); see Comments.

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A265801 Denominators of primes-only best approximates (POBAs) to the golden ratio, tau; see Comments.

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A265796 Numerators of lower primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

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A265797 Denominator of lower primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

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A265798 Numerators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

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A265799 Denominators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

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A265772 Numerators of lower primes-only best approximates (POBAs) to sqrt(2); see Comments.

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