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 13 results. Next

A007700 Numbers n such that n, 2n+1, and 4n+3 all prime.

Original entry on oeis.org

2, 5, 11, 41, 89, 179, 359, 509, 719, 1019, 1031, 1229, 1409, 1451, 1481, 1511, 1811, 1889, 1901, 1931, 2459, 2699, 2819, 3449, 3491, 3539, 3821, 3911, 5081, 5399, 5441, 5849, 6101, 6131, 6449, 7079, 7151, 7349, 7901, 8969, 9221, 10589, 10691, 10709, 11171
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

The corresponding primes 2n+1 and 4n+3 respectively have n-1 and 2n primitive roots. - Lekraj Beedassy, Jan 07 2005
At n > 2, a(n) == {11,29} (mod 30). - Zak Seidov, Jan 31 2013

References

  • T. Moreau, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Intersection of A005384 and A023213.
Cf. A005385, A023272, A023302, A023330, A057331, A005602.

Programs

  • Maple
    A007700 := proc(n) local p1,p2; p1 := 2*n+1; p2 := 2*p1+1; if isprime(n) = true and isprime(p1)=true and isprime(p2)=true then RETURN(n); fi; end;
  • Mathematica
    Select[Range[10^3*3], PrimeQ[ # ]&&PrimeQ[2*#+1]&&PrimeQ[4*#+3] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[1500]],AllTrue[{2#+1,4#+3},PrimeQ]&] (* Harvey P. Dale, Apr 12 2022 *)
  • PARI
    is(n)=isprime(n)&&isprime(2*n+1)&&isprime(4*n+3) \\ Charles R Greathouse IV, Mar 21 2013

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

A106015 Primes p such that 4*p +- 3 are primes.

Original entry on oeis.org

2, 5, 11, 19, 59, 89, 109, 149, 151, 331, 359, 389, 401, 439, 499, 521, 571, 599, 829, 941, 1019, 1039, 1129, 1249, 1279, 1319, 1381, 1451, 1669, 1741, 1871, 2131, 2161, 2179, 2251, 2459, 2819, 3119, 3251, 3469, 3539, 3581, 3659, 3911, 4001, 4231, 4261
Offset: 1

Views

Author

Zak Seidov, May 05 2005

Keywords

Links

Programs

  • Magma
    [p: p in PrimesUpTo(100000)| IsPrime(4*p-3) and IsPrime(4*p+3)]; // Vincenzo Librandi, Nov 13 2010
  • Mathematica
    Select[Prime[Range[400]], PrimeQ[4#+3]&&PrimeQ[4#-3]&]
Select[Prime[Range[600]],AllTrue[4#+{3,-3},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 18 2021 *)

Formula

A023213 INTERSECT A157978. - R. J. Mathar, Jul 25 2009

A157974 Primes p such that 12*p + 11 is also prime.

Original entry on oeis.org

3, 5, 13, 19, 29, 31, 41, 53, 59, 61, 71, 73, 101, 109, 113, 131, 151, 173, 199, 211, 223, 239, 241, 251, 263, 283, 293, 313, 389, 409, 419, 439, 449, 491, 503, 521, 523, 571, 593, 631, 641, 643, 659, 673, 701, 733, 769, 809, 811, 823, 839, 853, 883, 929, 953
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 10 2009

Keywords

Links

Crossrefs

Cf. A136082, A136083, A023213, A023221, A023235, A023240.

Programs

  • Magma
    [n: n in [0..1000] | IsPrime(n) and IsPrime(12*n + 11)]; // Vincenzo Librandi, Feb 03 2014
  • Mathematica
    q=11;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[1000]], PrimeQ[12 # + 11]&] (* Vincenzo Librandi, Feb 03 2014 *)

A157975 Primes p such that 16*p + 15 is also prime.

Original entry on oeis.org

2, 7, 11, 13, 23, 29, 37, 53, 61, 67, 71, 79, 89, 97, 103, 109, 113, 131, 137, 139, 149, 167, 179, 197, 211, 223, 257, 277, 293, 313, 317, 337, 379, 383, 397, 419, 431, 439, 443, 467, 571, 601, 617, 631, 641, 643, 653, 659, 677, 691, 719, 733, 739, 743, 809
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 10 2009

Keywords

Links

Crossrefs

Cf. A136082, A136083, A023213, A023221, A023235, A023240, A157974, A136083, A136084, A089440, A136085

Programs

  • Magma
    [n: n in [0..1000] | IsPrime(n) and IsPrime(16*n + 15)]; // Vincenzo Librandi, Feb 03 2014
  • Mathematica
    q=15;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[1000]], PrimeQ[16 # + 15]&] (* Vincenzo Librandi, Feb 03 2014 *)

A157978 Primes p such that 4*p - 3 is also a prime.

Original entry on oeis.org

2, 5, 11, 19, 23, 29, 59, 61, 71, 79, 89, 101, 103, 109, 113, 131, 149, 151, 191, 193, 233, 239, 263, 283, 313, 331, 353, 359, 373, 389, 401, 431, 439, 479, 499, 521, 523, 541, 569, 571, 599, 619, 631, 653, 659, 673, 683, 701, 709, 739, 743, 751, 761, 773, 829
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 10 2009

Keywords

Links

Crossrefs

Cf. A136082, A136083, A023213, A023221, A023235, A023240, A157974, A136083, A136084, A136085, A136086, A136087, A089440, A157975, A157976, A157977.

Programs

  • Magma
    [n: n in [0..2000] | IsPrime(n) and IsPrime(4*n - 3)]; // Vincenzo Librandi, Feb 03 2014
  • Mathematica
    q=3;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p-q],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[1000]],PrimeQ[4 # - 3]&] (* Vincenzo Librandi, Feb 03 2014 *)

A157976 Primes p such that 18*p + 17 is also prime.

Original entry on oeis.org

2, 3, 5, 13, 19, 23, 37, 47, 53, 67, 79, 83, 89, 103, 109, 149, 157, 167, 193, 229, 233, 257, 263, 277, 313, 347, 349, 383, 389, 419, 439, 457, 467, 487, 499, 523, 563, 569, 593, 599, 619, 677, 719, 727, 769, 773, 823, 829, 857, 863, 877, 937, 1013, 1039, 1049
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 10 2009

Keywords

Links

Crossrefs

Cf. A136082, A136083, A023213, A023221, A023235, A023240, A157974, A136083, A136084, A136085, A136086, A136087, A089440, A157975.

Programs

  • Magma
    [n: n in [0..1100] | IsPrime(n) and IsPrime(18*n + 17)]; // Vincenzo Librandi, Feb 03 2014
  • Mathematica
    q=17;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[1000]], PrimeQ[18 # + 17]&] (* Vincenzo Librandi, Feb 03 2014 *)

A157977 Primes p such that 20*p + 19 is also prime.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 41, 71, 101, 149, 167, 233, 239, 251, 263, 269, 281, 293, 317, 347, 353, 401, 449, 461, 491, 503, 557, 563, 569, 647, 683, 743, 797, 857, 941, 947, 953, 977, 1019, 1031, 1091, 1103, 1151, 1163, 1193, 1217, 1283, 1289, 1319, 1361, 1373
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 10 2009

Keywords

Links

Crossrefs

Cf. A136082, A136083, A023213, A023221, A023235, A023240, A157974, A136083, A136084, A136085, A136086, A136087, A089440, A157975, A157976.

Programs

  • Magma
    [n: n in [0..2000] | IsPrime(n) and IsPrime(20*n + 19)]; // Vincenzo Librandi, Feb 03 2014
  • Mathematica
    q=19;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[250]],PrimeQ[20#+19]&] (* Harvey P. Dale, Jul 04 2011 *)

A126330 Primes of the form 4p+3 where p is a prime.

Original entry on oeis.org

11, 23, 31, 47, 71, 79, 127, 151, 167, 191, 239, 271, 359, 431, 439, 599, 607, 631, 719, 727, 911, 919, 967, 1031, 1087, 1231, 1327, 1399, 1439, 1471, 1559, 1607, 1759, 1831, 1847, 1871, 1951, 1999, 2039, 2087, 2287, 2311, 2351, 2399, 2591, 2647, 2711, 2767
Offset: 1

Views

Author

J. M. Bergot, Mar 09 2007

Keywords

Links

Crossrefs

For the primes p see A023213.
Cf. A002145, A004767, A080148, A090866, A095278.

Programs

  • Maple
    select(p -> isprime(p) and isprime((p-3)/4), [seq(p,p=7..10000,4)]); # Robert Israel, Aug 08 2019
  • Mathematica
    Select[3 + 4Prime@Range[130], PrimeQ] (* Ray Chandler, Jun 29 2008 *)

Extensions

Checked and extended by N. J. A. Sloane, Mar 10 2007

A023281 Primes that remain prime through 3 iterations of function f(x) = 4x + 3.

Original entry on oeis.org

2, 109, 179, 571, 677, 977, 1279, 1447, 1747, 1901, 2207, 2671, 3119, 3917, 5011, 5399, 5441, 5569, 5791, 6211, 6607, 7079, 7417, 8369, 8831, 9221, 9697, 9769, 11821, 11897, 12347, 13537, 13669, 13691, 13729, 13781, 13907, 14747, 14851, 15581, 17231, 17497
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 4*p+3, 16*p+15 and 64*p+63 are also primes. - Vincenzo Librandi, Aug 04 2010

Links

Crossrefs

Subsequence of A023213, A023250, and of A095278.

Programs

  • Magma
    [n: n in [1..150000] | IsPrime(n) and IsPrime(4*n+3) and IsPrime(16*n+15) and IsPrime(64*n+63)] // Vincenzo Librandi, Aug 04 2010
  • Maple
    f:=proc(x) options operator, arrow: 4*x+3 end proc: a:=proc(n) if isprime(n)= true and isprime(f(n))=true and isprime(f(f(n)))=true and isprime(f(f(f(n)))) =true then n else end if end proc: seq(a(n),n=1..20000); # Emeric Deutsch, Jan 01 2008
  • Mathematica
    Select[Prime@ Range@ 2100, Times @@ Boole@ PrimeQ@ Rest@ NestList[4 # + 3 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 19 2016 *)
  • PARI
    is(n)=isprime(n) && isprime(4*n+3) && isprime(16*n+15) && isprime(64*n+63) \\ Charles R Greathouse IV, Sep 20 2016
Showing 1-10 of 13 results. Next

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

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