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 1-10 of 14 results. Next

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 *)

A265767 Numerators of upper primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

11, 37, 67, 97, 157, 307, 337, 367, 397, 487, 547, 757, 787, 907, 967, 997, 1117, 1567, 1657, 1747, 1867, 1987, 2287, 2437, 2617, 2707, 2857, 2887, 3037, 3067, 3217, 3307, 3457, 3547, 3637, 3697, 3847, 4057, 4297, 4597, 4957, 4987, 5107, 5167, 5197, 5347
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2015

Keywords

Comments

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.

Examples

			The upper POBAs to 5 start with 11/2, 37/7, 67/13, 97/19, 157/31, 307/61, 337/67, 367/73. For example, if p and q are primes and q > 19, and p/q > 5, then 97/19 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A023217, A158318, A265759, A265766, A222568, A265769.

Programs

  • Mathematica
    x = 5; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)

A265766 Numerators of lower primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

7, 13, 23, 53, 83, 113, 233, 263, 293, 353, 443, 503, 563, 653, 683, 743, 863, 953, 983, 1163, 1193, 1283, 1553, 1583, 1733, 1913, 2003, 2153, 2213, 2243, 2333, 2393, 2543, 2843, 2963, 3083, 3203, 3413, 3593, 3803, 3863, 4133, 4283, 4643, 4703, 4733, 5153
Offset: 1

Views

Author

Clark Kimberling, Dec 19 2015

Keywords

Comments

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.

Examples

			The lower POBAs to 5 start with 7/2, 13/3, 23/5, 53/11, 83/17, 113/23, 233/47. For example, if p and q are primes and q > 17, and p/q < 5, then 83/17 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A023217, A158318, A222568, A265759, A265767, A265769.

Programs

  • Mathematica
    x = 5; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)

A265768 Numerators of primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

7, 11, 23, 37, 53, 67, 83, 97, 113, 157, 233, 263, 293, 307, 337, 353, 367, 397, 443, 487, 503, 547, 563, 653, 683, 743, 757, 787, 863, 907, 953, 967, 983, 997, 1117, 1163, 1193, 1283, 1553, 1567, 1583, 1657, 1733, 1747, 1867, 1913, 1987, 2003, 2153, 2213
Offset: 1

Views

Author

Clark Kimberling, Dec 19 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 to 5 start with 7/2, 11/2, 23/5, 37/7, 53/11, 67/13, 83/17, 97/19, 113/23, 157/31, 233/47. For example, if p and q are primes and q > 13, then 67/13 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A023217, A158318, A265759, A265766, A265767, A265769.

Programs

  • Mathematica
    x = 5; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)

A265769 Denominators of primes-only best approximates (POBAs) to 5; see Comments.

Original entry on oeis.org

2, 2, 5, 7, 11, 13, 17, 19, 23, 31, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 109, 113, 131, 137, 149, 151, 157, 173, 181, 191, 193, 197, 199, 223, 233, 239, 257, 311, 313, 317, 331, 347, 349, 373, 383, 397, 401, 431, 443, 449, 457, 467, 479, 487, 509
Offset: 1

Views

Author

Clark Kimberling, Dec 20 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 to 5 start with 7/2, 11/2, 23/5, 37/7, 53/11, 67/13, 83/17, 97/19, 113/23, 157/31, 233/47. For example, if p and q are primes and q > 13, then 67/13 is closer to 5 than p/q is.

Crossrefs

Cf. A000040, A265759, A265766, A158318, A265767, A023217, A265768.

Programs

  • Mathematica
    x = 5; 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, A265768/A265769 *)
Numerator[tL]   (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU]   (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y]    (* A222568 *)
Denominator[y]  (* A265769 *)

A280720 For p = prime(n), number of iterations of the function f(x) = 5x + 2 that leave p prime.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Felix Fröhlich, Jan 07 2017

Keywords

Comments

Records are a(1) = 0 [p = 2], a(2) = 1 [p = 3], a(6) = 2 [p = 13], a(8) = 3 [p = 19], a(74) = 4 [p = 373], a(12656) = 6 [p = 135859], a(1165346) = 7 [p = 18235423], a(1659004) = 8 [p = 26588257], a(5386789) = 9 [p = 93112729], .... - Charles R Greathouse IV, Jan 12 2017

Links

Crossrefs

Cf. A016873, A023217, A023252, A023283, A023313, A023341, A084958, A249573.

Programs

  • Mathematica
    Table[Length@ NestWhileList[5 # + 2 &, Prime@ n, PrimeQ] - 2, {n, 120}] (* Michael De Vlieger, Jan 09 2017 *)
  • PARI
    a016873(n) = 5*n+2
a(n) = my(p=prime(n), i=0); while(1, if(!ispseudoprime(a016873(p)), return(i), p=a016873(p); i++))

A023283 Primes that remain prime through 3 iterations of function f(x) = 5x + 2.

Original entry on oeis.org

19, 373, 607, 1171, 1381, 1867, 2137, 2539, 3181, 4021, 5689, 5827, 5857, 6163, 7213, 7507, 11497, 12007, 13291, 13687, 14173, 15193, 16453, 21997, 22501, 24799, 25657, 28723, 31393, 31957, 32587, 35863, 40813, 42667, 42859, 43321, 43951, 45061, 45691
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 5*p+2, 25*p+12 and 125*p+62 are also primes. - Vincenzo Librandi, Aug 04 2010

Links

Crossrefs

Subsequence of A023217, A023252, and of A111223.

Programs

  • Magma
    [n: n in [1..150000] | IsPrime(n) and IsPrime(5*n+2) and IsPrime(25*n+12) and IsPrime(125*n+62)] // Vincenzo Librandi, Aug 04 2010
  • Mathematica
    Select[Prime@ Range@ 4800, Times @@ Boole@ PrimeQ@ Rest@ NestList[5 # + 2 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 20 2016 *)

Formula

a(n) == 1 (mod 6). - John Cerkan, Sep 20 2016

A023313 Primes that remain prime through 4 iterations of function f(x) = 5x + 2.

Original entry on oeis.org

373, 1171, 13687, 21997, 25657, 61603, 74413, 76471, 84199, 87181, 93487, 114691, 135859, 170761, 174877, 184333, 192979, 196177, 207931, 209743, 244219, 276229, 286687, 292561, 297811, 334603, 338893, 405037, 408361, 417097, 439141, 446323
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 5*p+2, 25*p+12, 125*p+62 and 625*p+312 are also primes. - Vincenzo Librandi, Aug 04 2010
Numbers k such that A280720(k) > 3. - Felix Fröhlich, Jan 07 2017

Links

Crossrefs

Cf. A280720. Subsequence of A023217, A023252, A023283, and A111223.

Programs

  • Magma
    [n: n in [1..1000000] | IsPrime(n) and IsPrime(5*n+2) and IsPrime(25*n+12) and IsPrime(125*n+62) and IsPrime(625*n+312)]; // Vincenzo Librandi, Aug 04 2010
  • Mathematica
    Select[Range[10^6], Times @@ Boole@ Map[PrimeQ, NestList[5 # + 2 &, #, 4]] == 1 &] (* Michael De Vlieger, Jan 09 2017 *)

Formula

a(n) == 31 or 37 (mod 42). - John Cerkan, Oct 07 2016

A023341 Primes that remain prime through 5 iterations of function f(x) = 5x + 2.

Original entry on oeis.org

135859, 174877, 192979, 244219, 292561, 679297, 842341, 964897, 1076029, 1470241, 1990579, 2004943, 2339263, 2615707, 2625577, 2633557, 2892277, 3003787, 3201901, 3758233, 4406797, 5065861, 5157547, 5390857, 5424961, 5546173, 5875369, 7746217
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 5*p+2, 25*p+12, 125*p+62, 625*p+312 and 3125*p+1562 are also primes. - Vincenzo Librandi, Aug 05 2010
Numbers k such that A280720(k) > 4. - Felix Fröhlich, Jan 07 2017

Links

Crossrefs

Subsequence of A023217, A023252, A023283, A023313, and A111223.
Cf. A280720.

Programs

  • Magma
    [n: n in [1..10000000] | IsPrime(n) and IsPrime(5*n+2) and IsPrime(25*n+12) and IsPrime(125*n+62) and IsPrime(625*n+312) and IsPrime(3125*n+1562)] // Vincenzo Librandi, Aug 05 2010
  • Mathematica
    p5Q[n_]:=And@@PrimeQ/@NestList[5#+2&,n,5]
Select[Prime[Range[550000]],p5Q]  (* Harvey P. Dale, Feb 17 2011 *)

Formula

a(n) == 31 (mod 42). - John Cerkan, Oct 17 2016

A249573 Smallest prime p that remains prime through exactly n iterations of the function f(x) = 5x + 2.

Original entry on oeis.org

2, 3, 13, 19, 373, 174877, 135859, 18235423, 26588257, 93112729, 376038903103, 7087694466289, 120223669028389
Offset: 0

Views

Author

Felix Fröhlich, Nov 01 2014

Keywords

Comments

Smallest p = prime(i) such that A280720(i) = n, where i is the index of p in A000040. - Felix Fröhlich, Jan 07 2017
From Jon E. Schoenfield, Jan 08 2017: (Start)
a(10) > 10^10.
It seems very likely that a(11) exists. But is it possible that this sequence is finite? Each row of the table below shows, for an interval of width 10^8, the number of primes p within the interval that remain prime through exactly 0 iterations, exactly 1 iteration, etc. E.g., in the interval [10^9, 10^9 + 10^8), there are 4437075 primes p that remain prime through exactly 0 iterations, 326699 that remain prime through exactly 1, 45062 that remain prime through exactly 2, etc.
---------------------------------------------------------------
Fixed interval width = 10^8
---------------------------------------------------------------
Start Number of successful iterations
of --------------------------------------------------------
intvl 0 1 2 3 4 5 6 7 8 9
===== ======= ====== ===== ===== ==== ==== ==== ==== ==== ====
1 5225638 450798 73434 10139 1308 114 17 4 2 1
10^ 8 4858227 391247 59352 7720 841 84 9 1 1 0
10^ 9 4437075 326699 45062 5438 605 45 10 2 0 0
10^10 4031707 271882 34218 3722 367 30 3 1 0 0
10^11 3689861 228960 26414 2649 251 20 6 0 0 0
10^12 3400459 194999 20675 1973 158 17 1 0 0 0
10^13 3155004 168786 16699 1489 108 6 1 0 0 0
10^14 2940881 147025 13535 1153 81 4 0 0 0 0
10^15 2752985 128743 11275 874 55 5 0 0 0 0
---------------------------------------------------------------
The numbers in column 0 drop at a rate that is not surprising, given that the way that the density of primes drops as numbers get larger. The numbers in the other columns drop more rapidly, in relative terms. Suppose a similar table were constructed using much wider intervals (perhaps with intervals starting not at 1, 10^8, 10^9, 10^10, etc., but at 1, 10^30, 10^31, 10^32, etc.), so that the numbers in, say, column 12 remained positive through several rows, but were dropping by a factor of more than 10 from one row to the next, making it likely that the total number of k-digit primes -- not just those from intervals of a fixed size -- that would remain prime through 12 iterations was actually decreasing as k increased. Would such an outcome suggest that the sequence might be finite? (End)

Examples

			With p = 13: 5 * 13 + 2 = 67, 5 * 67 + 2 = 337 and 5 * 337 + 2 = 1687. 67 and 337 are both prime, but 1687 is not, so 13 remains prime through exactly two iterations of 5 * x + 2 and is the smallest prime with this property, so a(2) = 13.

Crossrefs

Cf. A005602, A023217, A023313, A023341, A280720.

Programs

  • Mathematica
    c[p_] := Block[{k = 1, q = 5*p+2}, While[ PrimeQ[q], q = 5*q+2; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] != n, p = NextPrime@ p]; p]; Array[a, 7] (* Giovanni Resta, Mar 21 2017 *)
  • PARI
    for(n=0, 10, forprime(p=2, 1e20, i=0; a=p; while(ispseudoprime(5*a+2), a=5*a+2; i++); if(i==n, print1(p, ", "); break(1))))

Extensions

a(10) from Charles R Greathouse IV, Jan 13 2017
a(11) from John Cerkan, Mar 20 2017
a(12) from Giovanni Resta, Mar 21 2017
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.