A265759 -id:A265759 - cp's OEIS frontend

A265806 Numerators of primes-only best approximates (POBAs) to 1/(golden ratio) = 1/tau; see Comments.

2, 2, 3, 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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).

			The POBAs to 1/tau start with 2/2, 2/3, 3/5, 19/31, 23/37, 29/47, 97/157, 353/571. For example, if p and q are primes and q > 157, then 97/157 is closer to 1/tau than p/q is.

Cf. A000040, A265759, A265799, A265798, A265797, A265796, A265807.

x = 1/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, A265806/A265807 *)
Numerator[tL]   (* A265799 *)
Denominator[tL] (* A265798 *)
Numerator[tU]   (* A265797 *)
Denominator[tU] (* A265796 *)
Numerator[y]    (* A265806 *)
Denominator[y]  (* A265807 *)

a(14)-a(17) from Robert Price, Apr 06 2019

A265808 Numerators of lower primes-only best approximates (POBAs) to Pi; see Comments.

Original entry on oeis.org

5, 13, 19, 31, 37, 53, 97, 191, 223, 757, 977, 4483, 5237, 9497, 14423, 18061, 30841, 45751, 47881, 60661, 137341, 162901, 177811, 536273, 557573, 577453, 579583, 609403, 610823, 833719, 43354453, 45230587, 104426411, 111304859, 120059441, 185091653, 821656877, 1302520019

Offset: 1

Clark Kimberling, Jan 02 2016

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 Pi start with 5/2, 13/5, 19/7, 31/11, 37/13, 53/17, 97/31, 191/61, 223/71, 757/241, 977/311. For example, if p and q are primes and q > 241, and p/q < Pi, then 757/241 is closer to Pi than p/q is.

Cf. A000040, A265759, A265809, A265810, A265811, A265812, A265813.

x = Pi; 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, A265812/A265813 *)
Numerator[tL]   (* A265808 *)
Denominator[tL] (* A265809 *)
Numerator[tU]   (* A265810 *)
Denominator[tU] (* A265811 *)
Numerator[y]    (* A265812 *)
Denominator[y]  (* A265813 *)

More terms from Bert Dobbelaere, Jul 20 2022

A265809 Denominators of lower primes-only best approximates (POBAs) to Pi; see Comments.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 31, 61, 71, 241, 311, 1427, 1667, 3023, 4591, 5749, 9817, 14563, 15241, 19309, 43717, 51853, 56599, 170701, 177481, 183809, 184487, 193979, 194431, 265381, 13800151, 14397343, 33239959, 35429437, 38216107, 58916503, 261541507, 414604999, 549157573

Offset: 1

Clark Kimberling, Jan 02 2016

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 Pi start with 5/2, 13/5, 19/7, 31/11, 37/13, 53/17, 97/31, 191/61, 223/71, 757/241, 977/311. For example, if p and q are primes and q > 241, and p/q < Pi, then 757/241 is closer to Pi than p/q is.

Cf. A000040, A265759, A265808, A265810, A265811, A265812, A265813.

x = Pi; 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, A265812/A265813 *)
Numerator[tL]   (* A265808 *)
Denominator[tL] (* A265809 *)
Numerator[tU]   (* A265810 *)
Denominator[tU] (* A265811 *)
Numerator[y]    (* A265812 *)
Denominator[y]  (* A265813 *)

More terms from Bert Dobbelaere, Jul 20 2022

A265810 Numerators of upper primes-only best approximates (POBAs) to Pi; see Comments.

Original entry on oeis.org

7, 17, 23, 41, 167, 211, 431, 563, 569, 619, 1109, 5413, 10427, 16063, 20323, 28843, 47969, 56489, 71399, 75659, 99089, 107609, 118259, 145949, 203459, 211891, 626443, 1668503, 5628757, 16259473, 20011031, 38144863, 40436969, 72536393, 93379723, 114431749, 136109153, 266173577

Offset: 1

Clark Kimberling, Jan 02 2016

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 Pi start with 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 431/137. For example, if p and q are primes and q > 67, and p/q > Pi, then 211/67 is closer to Pi than p/q is.

Cf. A000040, A265759, A265808, A265809, A265811, A265812, A265813.

x = Pi; 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, A265812/A265813 *)
Numerator[tL]   (* A265808 *)
Denominator[tL] (* A265809 *)
Numerator[tU]   (* A265810 *)
Denominator[tU] (* A265811 *)
Numerator[y]    (* A265812 *)
Denominator[y]  (* A265813 *)

More terms from Bert Dobbelaere, Jul 20 2022

A265811 Denominators of upper primes-only best approximates (POBAs) to Pi; see Comments.

Original entry on oeis.org

2, 5, 7, 13, 53, 67, 137, 179, 181, 197, 353, 1723, 3319, 5113, 6469, 9181, 15269, 17981, 22727, 24083, 31541, 34253, 37643, 46457, 64763, 67447, 199403, 531101, 1791689, 5175551, 6369709, 12141887, 12871487, 23089051, 29723689, 36424757, 43324889, 84725681, 105426077, 110667493

Offset: 1

Clark Kimberling, Jan 02 2016

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 Pi start with 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 431/137. For example, if p and q are primes and q > 67, and p/q > Pi, then 211/67 is closer to Pi than p/q is.

Cf. A000040, A265759, A265808, A265809, A265810, A265812, A265813.

x = Pi; 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, A265812/A265813 *)
Numerator[tL]   (* A265808 *)
Denominator[tL] (* A265809 *)
Numerator[tU]   (* A265810 *)
Denominator[tU] (* A265811 *)
Numerator[y]    (* A265812 *)
Denominator[y]  (* A265813 *)

More terms from Bert Dobbelaere, Jul 20 2022

A265812 Numerators of primes-only best approximates (POBAs) to Pi; see Comments.

Original entry on oeis.org

5, 7, 17, 23, 41, 167, 211, 223, 619, 757, 977, 1109, 4483, 5237, 5413, 9497, 14423, 16063, 18061, 30841, 45751, 47881, 60661, 137341, 162901, 177811, 211891, 626443, 833719, 38144863, 40436969, 45230587, 93379723, 114431749, 120059441, 185091653, 347672183, 1725229397, 1736068099

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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).

			The POBAs to Pi start with 5/2, 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 223/71, 619/197. For example, if p and q are primes and q < 53, then 167/53 is closer to Pi than p/q is.

Cf. A000040, A265759, A265808, A265809, A265810, A265811, A265813.

x = Pi; 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, A265812/A265813 *)
Numerator[tL]   (* A265808 *)
Denominator[tL] (* A265809 *)
Numerator[tU]   (* A265810 *)
Denominator[tU] (* A265811 *)
Numerator[y]    (* A265812 *)
Denominator[y]  (* A265813 *)

More terms from Bert Dobbelaere, Jul 20 2022

A265813 Denominators of primes-only best approximates (POBAs) to Pi; see Comments.

Original entry on oeis.org

2, 2, 5, 7, 13, 53, 67, 71, 197, 241, 311, 353, 1427, 1667, 1723, 3023, 4591, 5113, 5749, 9817, 14563, 15241, 19309, 43717, 51853, 56599, 67447, 199403, 265381, 12141887, 12871487, 14397343, 29723689, 36424757, 38216107, 58916503, 110667493, 549157573, 552607639

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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).

			The POBAs to Pi start with 5/2, 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 223/71, 619/197. For example, if p and q are primes and q < 53, then 167/53 is closer to Pi than p/q is.

Cf. A000040, A265759, A265808, A265809, A265810, A265811, A265812.

x = Pi; 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, A265812/A265813 *)
Numerator[tL]   (* A265808 *)
Denominator[tL] (* A265809 *)
Numerator[tU]   (* A265810 *)
Denominator[tU] (* A265811 *)
Numerator[y]    (* A265812 *)
Denominator[y]  (* A265813 *)

More terms from Bert Dobbelaere, Jul 20 2022

A265814 Numerators of lower primes-only best approximates (POBAs) to e; see Comments.

Original entry on oeis.org

5, 13, 19, 307, 443, 617, 2237, 2411, 2971, 5923, 7043, 7603, 11887, 12659, 15361, 24103, 75223, 89021, 128273, 283949, 423299, 1169027, 1587077, 1830211, 3062207, 5080939, 8481901, 9366979, 22675801, 67090433, 71625049, 191016521, 211670869, 221578729, 244402043, 428023867, 1451377009

Offset: 1

Clark Kimberling, Jan 02 2016

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 e; start with 5/2, 13/5, 19/7, 307/113, 443/163, 617/227, 2237/823. For example, if p and q are primes and q > 823, and p/q < e, then 2237/823 is closer to e than p/q is.

Cf. A000040, A265759, A265815, A265816, A265817, A265818, A265819.

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 *)

More terms from Bert Dobbelaere, Jul 21 2022

A265815 Denominators of lower primes-only best approximates (POBAs) to e; see Comments.

Original entry on oeis.org

2, 5, 7, 113, 163, 227, 823, 887, 1093, 2179, 2591, 2797, 4373, 4657, 5651, 8867, 27673, 32749, 47189, 104459, 155723, 430061, 583853, 673297, 1126523, 1869173, 3120317, 3445919, 8341961, 24681191, 26349383, 70271051, 77869361, 81514259, 89910487, 157461181, 533931763, 583892083, 770930497

Offset: 1

Clark Kimberling, Jan 02 2016

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 e; start with 5/2, 13/5, 19/7, 307/113, 443/163, 617/227, 2237/823. For example, if p and q are primes and q > 823, and p/q < e, then 2237/823 is closer to e than p/q is.

Cf. A000040, A265759, A265814, A265816, A265817, A265818, A265819.

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 *)

More terms from Bert Dobbelaere, Jul 21 2022

A265816 Numerators of upper primes-only best approximates (POBAs) to e; see Comments.

Original entry on oeis.org

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

Clark Kimberling, Jan 02 2016

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 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.

Cf. A000040, A265759, A265814, A265815, A265817, A265818, A265819.

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 *)

More terms from Bert Dobbelaere, Jul 21 2022

cp's OEIS Frontend

A265806 Numerators of primes-only best approximates (POBAs) to 1/(golden ratio) = 1/tau; see Comments.

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A265808 Numerators of lower primes-only best approximates (POBAs) to Pi; see Comments.

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A265809 Denominators of lower primes-only best approximates (POBAs) to Pi; see Comments.

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A265810 Numerators of upper primes-only best approximates (POBAs) to Pi; see Comments.

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A265811 Denominators of upper primes-only best approximates (POBAs) to Pi; see Comments.

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A265812 Numerators of primes-only best approximates (POBAs) to Pi; see Comments.

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A265813 Denominators of primes-only best approximates (POBAs) to Pi; see Comments.

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A265814 Numerators of lower primes-only best approximates (POBAs) to e; see Comments.

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