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 *)

A265761 Numerators of primes-only best approximates (POBAs) to 3/2; see Comments.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 43, 47, 61, 71, 79, 89, 101, 107, 109, 151, 163, 191, 197, 223, 227, 251, 269, 271, 317, 349, 359, 421, 439, 461, 467, 521, 523, 569, 601, 613, 631, 647, 659, 673, 691, 701, 719, 811, 821, 853, 857, 881, 911, 919, 929, 947, 971, 991
Offset: 1

Views

Author

Clark Kimberling, Dec 18 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 for 3/2 start with 2/2, 5/3, 7/5, 11/7, 17/11, 19/13, 29/19, 43/29, 47/31. For example, if p and q are primes and q > 13, then 19/13 is closer to 3/2 than p/q is.

Crossrefs

Cf. A000040, A104163, A158708, A158790, A162336, A222565.

Programs

  • Mathematica
    x = 3/2; 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, A265761/A222565 *)
Numerator[tL]   (* A104163 *)
Denominator[tL] (* A158708 *)
Numerator[tU]   (* A162336 *)
Denominator[tU] (* A158709 *)
Numerator[y]    (* A265761 *)
Denominator[y]  (* A222565 *)

A297306 Primes p such that q = 4*p+1 and r = (2*p+1)/3 are also primes.

Original entry on oeis.org

7, 43, 79, 163, 673, 853, 919, 1063, 1429, 1549, 1663, 2143, 2683, 3229, 3499, 4993, 5119, 5653, 5779, 6229, 6343, 7333, 7459, 7669, 8353, 8539, 8719, 9829, 10009, 10243, 10303, 11383, 11689, 12583, 13399, 14149, 14653, 14923, 15649, 16603, 17053, 17389, 17749
Offset: 1

Views

Author

David S. Newman, Jan 04 2018

Keywords

Comments

This sequence was suggested by Moshe Shmuel Newman. It has its source in his study of finite groups.

Examples

			Prime p = 7 is in the sequence because q = 4*7+1 = 29 and r = (2*7+1)/3 = 5 are also primes.

Links

Crossrefs

Cf. A000040.
Intersection of A023212 and A104163.

Programs

  • Maple
    a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p); if irem(p, 3)=1 and
         isprime(4*p+1) and isprime((2*p+1)/3) then break fi
      od; p
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 07 2018
  • Mathematica
    a[n_] := a[n] = Module[{p}, p = If[n == 1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[Mod[p, 3] == 1 && PrimeQ[4p+1] && PrimeQ[(2p+1)/3], Break[]]]; p];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)
  • PARI
    isok(p) = isprime(p) && isprime(4*p+1) && iferr(isprime((2*p+1)/3), E, 0); \\ Michel Marcus, Nov 27 2020

Extensions

More terms from Alois P. Heinz, Jan 07 2018

A321510 Primes p for which there exists a prime q < p such that 3*q == 1 (mod p).

Original entry on oeis.org

5, 7, 19, 43, 61, 79, 109, 151, 163, 223, 271, 349, 421, 439, 523, 601, 613, 631, 673, 691, 811, 853, 919, 991, 1009, 1051, 1063, 1153, 1213, 1231, 1279, 1321, 1429, 1531, 1549, 1663, 1693, 1789, 1801, 1873, 1933, 1951, 2113, 2143, 2179, 2221, 2239, 2503, 2539, 2683, 2791, 2833, 2851
Offset: 1

Views

Author

David James Sycamore, Nov 11 2018

Keywords

Comments

A104163 with 5 prepended (see example). For any prime p in A104163 q = (2*p+1)/3, then q < p and 3*q == 1 (mod p).

Examples

			For p = 11, the only number t < 11 such that 3*t == 1 (mod 11) is t = 4, which is not prime, therefore 11 is not a term.
For p = 5, q = 2 (prime); 2*3 = 6 == 1 (mod 5) therefore 5 is a term.

Crossrefs

Cf. A104163 (essentially the same sequence), A005383.

Programs

  • Maple
    for n from 3 to 300 do
Y := ithprime(n);
Z := 1/3 mod Y;
if isprime(Z) then print(Y);
end if:
end do:
  • Mathematica
    aQ[p_]:=Module[{ans=False, q=2}, While[qAmiram Eldar, Nov 12 2018 *)
Join[{5}, Select[Prime[Range[400]], PrimeQ[((2 # + 1)) / 3] &]] (* Vincenzo Librandi, Nov 17 2018 *)
  • PARI
    isok(p) = if (isprime(p), forprime(q=1, p-1, if ((3*q % p) == 1, return (1)))); \\ Michel Marcus, Nov 14 2018

Formula

a(n+1) = A104163(n); n >= 1.
Showing 1-4 of 4 results.

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