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 41 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 *)

A050703 Numbers that when added to the sum of their prime factors (with multiplicity) become prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 26, 33, 34, 35, 38, 44, 46, 48, 51, 55, 57, 58, 65, 68, 74, 85, 86, 90, 93, 96, 111, 112, 116, 118, 123, 135, 141, 143, 145, 155, 158, 161, 166, 177, 178, 185, 188, 194, 201, 203, 205, 206, 208, 209, 210, 212, 215, 221, 224, 225, 252
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Comments

No term of this sequence can be prime, since for a prime p, A075254(p)=2*p, hence not prime. - Michel Marcus, Jul 24 2015
From Robert Israel, Jul 24 2015: (Start)
Similarly, no term of the sequence can be a prime power.
Contains 2*n for n in A023208 and 3*n for n in A023213. (End)

Examples

			252 = 2*2*3*3*7; 252 + (2 + 2 + 3 + 3 + 7) = 252 + 17 = 269, which is prime.

Links

Crossrefs

Cf. A050704-A050710, A075254, A023208, A023213.

Programs

  • Maple
    filter:= n ->isprime(convert(map(convert,ifactors(n)[2],`*`),`+`)+n):
select(filter, [$1..1000]); # Robert Israel, Jul 24 2015
  • Mathematica
    upto=300;Rest[Select[Complement[Range[upto], Prime[Range[ PrimePi[upto]]]], PrimeQ[#+ Total[Times@@@FactorInteger[#]]]&]] (* Harvey P. Dale, Apr 20 2011 *)
Select[Range[500], PrimeQ[# + Total [Times @@@ FactorInteger[#]] && PrimeOmega[#] > 1] &]  (* K. D. Bajpai, Sep 12 2014 *)
  • PARI
    sopfr(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1]*f[i,2])
is(n)=!isprime(n)&&isprime(n+sopfr(n)) \\ Charles R Greathouse IV, Jul 19 2011

Formula

{n: A075254(n) in A000040}. - R. J. Mathar, Jul 27 2015

Extensions

Name clarified by Michel Marcus, Jul 24 2015

A094524 Primes of form 3*prime(m) + 2.

Original entry on oeis.org

11, 17, 23, 41, 53, 59, 71, 89, 113, 131, 179, 239, 251, 269, 293, 311, 383, 419, 449, 491, 503, 521, 593, 599, 683, 701, 719, 773, 809, 881, 941, 953, 1013, 1049, 1061, 1103, 1151, 1193, 1229, 1259, 1301, 1319, 1373, 1439, 1499, 1511, 1571, 1709, 1733
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2004

Keywords

Examples

			a(7) = 3*23 + 2 = 71.

Links

Crossrefs

Cf. A023208, A091180.

Programs

Formula

a(n) = 3*A023208(n) + 2.

A125272 Primes p such that 3p - 2 and 3p + 2 are also primes.

Original entry on oeis.org

3, 5, 7, 13, 23, 37, 43, 103, 127, 163, 167, 257, 293, 313, 337, 433, 523, 757, 797, 887, 953, 1013, 1063, 1153, 1283, 1303, 1307, 1483, 1597, 1657, 1667, 1693, 1723, 1783, 1913, 2003, 2333, 2347, 2557, 2897, 2927, 3067, 3533, 3823, 3943, 4003, 4013, 4093
Offset: 1

Views

Author

Zak Seidov, Nov 26 2006

Keywords

Links

Crossrefs

Intersection of A023208 and A088878.
Cf. A125215.

Programs

  • Magma
    [p: p in PrimesUpTo(70000)| IsPrime(3*p-2)and IsPrime(3*p+2)] // Vincenzo Librandi, Jan 29 2011
  • Mathematica
    lst={}; Do[p=Prime[n]; If[PrimeQ[3*p-2]&&PrimeQ[3*p+2],AppendTo[lst,p]],{n,7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 18 2009 *)
Select[Prime[Range[600]],AllTrue[3#+{2,-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2021 *)
  • PARI
    is(n)=isprime(3*n-2)&&isprime(3*n+2)&&isprime(n) \\ Charles R Greathouse IV, Jul 02 2013

Formula

a(n) = A125215(n)/3.

A136082 Son primes of order 5.

Original entry on oeis.org

3, 11, 17, 23, 41, 53, 59, 107, 131, 167, 173, 179, 191, 257, 263, 269, 389, 401, 431, 461, 467, 479, 521, 563, 569, 599, 647, 653, 677, 683, 719, 773, 821, 839, 857, 887, 947, 971, 1031, 1049, 1061, 1091, 1103, 1151, 1181, 1217, 1223, 1259, 1277, 1301
Offset: 1

Views

Author

Artur Jasinski, Dec 12 2007

Keywords

Comments

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235.
Numbers in this sequence are those primes p such that 11*p + 10 is also prime. Generally, son primes of order n are the primes p such that (2n+1)*p + 2n is also prime. - Bob Selcoe, Apr 04 2015

Links

Crossrefs

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136083, A136084, A136085, A136086, A136087, A136088, A136089, A136090, A136091.

Programs

  • Mathematica
    n = 5; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=10;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
Select[Prime[Range[250]],PrimeQ[11#+10]&] (* Harvey P. Dale, Aug 07 2021 *)

A136083 Son primes of order 6.

Original entry on oeis.org

7, 13, 17, 23, 29, 43, 53, 67, 79, 83, 109, 113, 127, 149, 157, 163, 179, 193, 227, 233, 239, 277, 283, 293, 307, 317, 347, 349, 359, 367, 373, 433, 449, 457, 487, 503, 557, 563, 587, 619, 647, 653, 673, 727, 739, 769, 773, 787, 809, 823, 829, 883, 919, 947
Offset: 1

Views

Author

Artur Jasinski, Dec 12 2007

Keywords

Comments

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235. For son primes of order 5 see A136082.

Links

Crossrefs

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136082, A136084, A136085, A136086, A136087, A136088, A136089, A136090, A136091.

Programs

  • Mathematica
    n = 6; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=12;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)

A136084 Son primes of order 7.

Original entry on oeis.org

3, 5, 11, 17, 23, 29, 31, 37, 43, 47, 53, 61, 67, 73, 83, 103, 107, 113, 131, 137, 139, 163, 173, 179, 181, 191, 193, 197, 199, 223, 229, 251, 269, 271, 281, 283, 293, 311, 353, 359, 367, 389, 401, 419, 421, 439, 443, 457, 463, 467, 499, 503, 509, 521, 547, 557
Offset: 1

Views

Author

Artur Jasinski, Dec 12 2007

Keywords

Comments

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235. For son primes of order 5 see A136082. For son primes of order 6 see A136083.

Links

Crossrefs

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136082, A136083, A136085, A136086, A136087, A136088, A136089, A136090, A136091.

Programs

  • Mathematica
    n = 7; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=14;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)

A136085 Son primes of order 8.

Original entry on oeis.org

3, 5, 29, 59, 71, 83, 101, 131, 149, 173, 239, 251, 281, 311, 401, 443, 449, 461, 491, 509, 563, 569, 653, 701, 719, 743, 761, 929, 953, 1109, 1151, 1193, 1223, 1259, 1289, 1301, 1373, 1451, 1511, 1553, 1571, 1583, 1613, 1619, 1811, 1913, 1931, 1949, 2039
Offset: 1

Views

Author

Artur Jasinski, Dec 12 2007

Keywords

Comments

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235. For son primes of order 5 see A136082. For son primes of order 6 see A136083. For son primes of order 7 see A136084.

Links

Crossrefs

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136082, A136083, A136084, A136086, A136087, A136088, A136089, A136090, A136091.

Programs

  • Mathematica
    n = 8; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=16;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)

A136086 Son primes of order 9.

Original entry on oeis.org

5, 7, 11, 19, 29, 31, 41, 47, 67, 71, 89, 97, 109, 137, 139, 151, 157, 167, 181, 197, 211, 241, 251, 271, 277, 307, 311, 337, 367, 379, 397, 409, 421, 509, 557, 571, 587, 599, 601, 607, 619, 631, 641, 659, 661, 691, 701, 719, 727, 757, 769, 797, 811, 827, 839
Offset: 1

Views

Author

Artur Jasinski, Dec 12 2007

Keywords

Comments

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235. For son primes of order 5 see A136082. For son primes of order 6 see A136083. For son primes of order 7 see A136084. For son primes of order 8 see A136085.

Links

Crossrefs

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136082, A136083, A136084, A136085, A136087, A136088, A136089, A136090, A136091.

Programs

  • Mathematica
    n = 9; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=18;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)

A136087 Son primes of order 10.

Original entry on oeis.org

3, 7, 11, 13, 19, 23, 37, 41, 59, 61, 67, 71, 73, 89, 101, 107, 109, 113, 127, 137, 139, 151, 167, 179, 181, 193, 197, 211, 223, 227, 239, 241, 257, 269, 271, 293, 311, 331, 347, 349, 353, 359, 367, 373, 409, 419, 421, 439, 443, 463, 479, 487, 491, 499, 509
Offset: 1

Views

Author

Artur Jasinski, Dec 12 2007

Keywords

Comments

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235. For son primes of order 5 see A136082. For son primes of order 6 see A136083. For son primes of order 7 see A136084. For son primes of order 8 see A136085. For son primes of order 8 see A136086.

Links

Crossrefs

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136082, A136083, A136084, A136085, A136086, A136088, A136089, A136090, A136091.

Programs

  • Mathematica
    n = 10; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=20;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
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