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.

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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 *)

A349666 Primes of the form 4*k+3 that are still a prime of the form 4*k+3 after 2 Collatz steps.

Original entry on oeis.org

7, 31, 47, 71, 127, 151, 167, 239, 311, 431, 439, 479, 607, 631, 647, 727, 839, 911, 967, 991, 1039, 1231, 1319, 1399, 1471, 1511, 1559, 1567, 1607, 1879, 1951, 1999, 2111, 2239, 2311, 2351, 2447, 2671, 2719, 2927, 3119, 3167, 3191, 3359, 3391, 3671, 3727, 3767, 3911
Offset: 1

Views

Author

Karl-Heinz Hofmann, Dec 23 2021

Keywords

Comments

The two Collatz steps are 3*x + 1 and x/2.
Terms are primes in A002145 which after 2 Collatz iterations are still a prime in A002145.
Pythagorean primes (A002144), which are of the form 4*k+1, never produce any prime after those 2 steps. But further reducing by 2 produces primes in A349667.
Apparently this is a subsequence of A158709.

Examples

			(31*3 + 1)/2 = 47. Both 31 and 47 are primes of the form 4*k+3. Thus 31 is a term.

Links

Crossrefs

Cf. A002145, A002144, A158709, A349667.

Programs

  • Mathematica
    Select[4*Range[0, 1000] + 3, PrimeQ[#] && Mod[(q = (3*# + 1)/2), 4] == 3 && PrimeQ[q] &] (* Amiram Eldar, Dec 24 2021 *)
  • PARI
    isok(p) = isprime(p) && ((p%4)==3) && isprime(q=(3*p+1)/2) && ((q%4)==3); \\ Michel Marcus, Dec 23 2021

Formula

a(n) == 7 (mod 8). - Hugo Pfoertner, Dec 25 2021

A158722 Primes p which are not in A158720 and A158721.

Original entry on oeis.org

3, 7, 11, 19, 29, 37, 41, 43, 47, 61, 71, 79, 83, 89, 97, 101, 107, 109, 127, 131, 137, 139, 151, 157, 163, 173, 191, 199, 223, 227, 229, 239, 241, 251, 257, 263, 271, 277, 281, 283, 293, 311, 313, 317, 331, 349, 353, 367, 373, 379, 383, 389, 397, 401, 409, 419
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714, A158719, A158720, A158721

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[ !PrimeQ[Floor[p/3]+p]&&!PrimeQ[Ceiling[p/3]+p],AppendTo[lst,p]],{n,5!}];lst

A158723 Greater of twin primes in A158720.

Original entry on oeis.org

13, 31, 73, 103, 181, 193, 433, 463, 571, 643, 661, 823, 1021, 1291, 1621, 1723, 2083, 2143, 2341, 2593, 2713, 3001, 3253, 3331, 3361, 3541, 4231, 4243, 4423, 4933, 5233, 5653, 5881, 6553, 6571, 6781, 6871, 6961, 7951, 8293, 9283, 9343, 9433, 9631, 9931
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 24 2009

Keywords

Comments

If prime number from sequence A158720 is twin prime, it always (?) Greater of twin primes, and none (?) of Lesser of twin primes.

Crossrefs

Cf. A158708, A158709, A158710, A158711, A158712, A158713, A158714, A158719, A158720, A158721, A158722

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[Floor[p/3]+p],If[PrimeQ[p-2],AppendTo[lst,p]]],{n,7!}];lst
Previous Showing 11-14 of 14 results.

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