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 12 results. Next

A127581 Smallest prime of the form k*2^n - 1, for k >= 2.

Original entry on oeis.org

2, 3, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727, 81919, 131071, 131071, 524287, 524287, 14680063, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647
Offset: 0

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Examples

			a(3)=23 because 23 = 3*2^3 - 1 is prime.
a(4)=31 because 31 = 2*2^4 - 1 is prime.

Crossrefs

Cf. A007522, A127575-A127582, A127586-A127587.

Programs

  • Mathematica
    a = {}; Do[k = 1; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 0, 50}]; a

Formula

a(n) << 37^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Extensions

Edited by Don Reble, Jun 11 2007

A127586 Smallest strictly positive integer k such that (k+1)*2^n-1 is prime.

Original entry on oeis.org

2, 1, 1, 2, 1, 3, 1, 2, 4, 6, 4, 2, 1, 8, 4, 3, 1, 3, 1, 27, 13, 6, 25, 12, 38, 21, 10, 15, 7, 3, 1, 9, 4, 5, 2, 23, 11, 5, 2, 24, 23, 11, 5, 2, 13, 6, 19, 9, 4, 18, 10
Offset: 0

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Comments

The associated prime number list is (k+1)*2^n-1 = 2, 3, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727 for n=0,1,2,3,4,... - R. J. Mathar, Jan 22 2007

Crossrefs

Cf. A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127587.

Programs

  • Maple
    A127586 := proc(n) local k; k:=1 ; while true do if isprime( (k+1)*2^n-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 0 to 100 do printf("%d, ",A127586(n)) ; od ; # R. J. Mathar, Jan 22 2007
  • Mathematica
    a = {}; Do[k = 1; While[ ! PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k], {n, 0, 50}]; a

Formula

a(n)=A127587(n) if n is not in A000043. - R. J. Mathar, Jan 22 2007
a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

A127589 Primes of the form 16k + 5.

Original entry on oeis.org

5, 37, 53, 101, 149, 181, 197, 229, 277, 293, 373, 389, 421, 613, 661, 677, 709, 757, 773, 821, 853, 997, 1013, 1061, 1093, 1109, 1237, 1301, 1381, 1429, 1493, 1621, 1637, 1669, 1733, 1861, 1877, 1973, 2053, 2069, 2213, 2293, 2309, 2341, 2357, 2389, 2437
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Comments

All terms are the sum of two squares.
Primes with least significant digit 5 in hexadecimal. - Alonso del Arte, Oct 21 2022

Links

Crossrefs

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[16n + 5], AppendTo[a, 16n + 5]], {n, 0, 200}]; a
Select[16Range[200] + 5, PrimeQ] (* Alonso del Arte, Oct 21 2022 *)
  • PARI
    select(x->(x%16)==5, primes(500)) \\ Michel Marcus, Oct 24 2022

Extensions

Invalid comment removed by Zak Seidov, Jul 22 2010

A035089 Smallest prime of form 2^n*k + 1.

Original entry on oeis.org

2, 3, 5, 17, 17, 97, 193, 257, 257, 7681, 12289, 12289, 12289, 40961, 65537, 65537, 65537, 786433, 786433, 5767169, 7340033, 23068673, 104857601, 167772161, 167772161, 167772161, 469762049, 2013265921, 3221225473, 3221225473, 3221225473, 75161927681
Offset: 0

Views

Author

Labos Elemer

Keywords

Comments

a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order 2^n. - Joerg Arndt, Oct 18 2020

Links

Crossrefs

Analogous case is A034694. Fermat primes (A019434) are a subset. See also Fermat numbers A000215.
Cf. A007522, A057775, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

Programs

  • Mathematica
    a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k 2^n + 1], {n, 1, 50}]; a (* Artur Jasinski *)
  • PARI
    a(n)=for(k=1,9e99,if(ispseudoprime(k<Charles R Greathouse IV, Jul 06 2011

Extensions

a(0) from Joerg Arndt, Jul 06 2011

A127590 Numbers n such that 16n+5 is prime.

Original entry on oeis.org

0, 2, 3, 6, 9, 11, 12, 14, 17, 18, 23, 24, 26, 38, 41, 42, 44, 47, 48, 51, 53, 62, 63, 66, 68, 69, 77, 81, 86, 89, 93, 101, 102, 104, 108, 116, 117, 123, 128, 129, 138, 143, 144, 146, 147, 149, 152, 159, 167, 168, 171, 174, 177, 182, 191, 194
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Links

Crossrefs

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[16n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[0,200],PrimeQ[16#+5]&] (* Harvey P. Dale, Aug 31 2020 *)
  • PARI
    is(n)=isprime(16*n+5) \\ Charles R Greathouse IV, Feb 17 2017

A127591 Numbers k such that 64k+21 is prime.

Original entry on oeis.org

2, 4, 10, 13, 17, 19, 20, 22, 23, 25, 29, 32, 37, 44, 50, 53, 55, 58, 59, 62, 68, 79, 83, 88, 89, 94, 95, 97, 100, 107, 109, 113, 118, 122, 134, 142, 143, 152, 155, 157, 158, 163, 167, 169, 173, 193, 194, 199, 200
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Links

Crossrefs

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127592, A127593, A127594.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[21 + 64 n], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[200],PrimeQ[64#+21]&] (* Harvey P. Dale, Jan 15 2016 *)

A127592 Primes of the form 64k+21.

Original entry on oeis.org

149, 277, 661, 853, 1109, 1237, 1301, 1429, 1493, 1621, 1877, 2069, 2389, 2837, 3221, 3413, 3541, 3733, 3797, 3989, 4373, 5077, 5333, 5653, 5717, 6037, 6101, 6229, 6421, 6869, 6997, 7253, 7573, 7829, 8597, 9109, 9173, 9749, 9941, 10069, 10133, 10453
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007, Nov 12 2007

Keywords

Comments

All these primes are sums of two squares, also all indices are sums of two squares since we have the identity 64k+21 = 4(4(4k+1)+1)+1.

Links

Crossrefs

Cf. A000040. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127593, A127594.

Programs

  • Magma
    [p: p in PrimesUpTo(11000) | p mod 64 eq 21 ]; // Vincenzo Librandi, Sep 06 2012
  • Mathematica
    a = {}; Do[If[PrimeQ[21 + 64 n], AppendTo[a, 21 + 64 n]], {n, 0, 200}]; a
Select[Prime[Range[1700]], MemberQ[{21}, Mod[#, 64]] &] (* Vincenzo Librandi, Sep 06 2012 *)

A127593 Primes of the form 256 k + 85.

Original entry on oeis.org

853, 1109, 1621, 1877, 2389, 3413, 5717, 6229, 6997, 7253, 10069, 10837, 11093, 12373, 13397, 16981, 17749, 18517, 18773, 19541, 21589, 22613, 23893, 24917, 27733, 29269, 30293, 31573, 32341, 37717, 39509, 40277, 41813, 43093, 46933
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Crossrefs

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127594.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[85 + 256 n], AppendTo[a, 85 + 256 n]], {n, 0, 200}]; a
Select[256*Range[200]+85,PrimeQ] (* Harvey P. Dale, Oct 09 2020 *)

A127594 Numbers k such that 256 k + 85 is prime.

Original entry on oeis.org

3, 4, 6, 7, 9, 13, 22, 24, 27, 28, 39, 42, 43, 48, 52, 66, 69, 72, 73, 76, 84, 88, 93, 97, 108, 114, 118, 123, 126, 147, 154, 157, 163, 168, 183, 184, 186, 196, 198
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Crossrefs

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593.

Programs

  • Mathematica
    a = {}; Do[If[PrimeQ[85 + 256 n], AppendTo[a, n]], {n, 0, 200}]; a

A127582 a(n) = the smallest prime number of the form k*2^n - 1, for k >= 1.

Original entry on oeis.org

2, 3, 3, 7, 31, 31, 127, 127, 1279, 3583, 5119, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647
Offset: 0

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Examples

			a(0)=2 because 2 = 3*2^0 - 1 is prime.
a(1)=3 because 3 = 2*2^1 - 1 is prime.
a(2)=3 because 3 = 1*2^2 - 1 is prime.
a(3)=7 because 7 = 1*2^3 - 1 is prime.
a(4)=31 because 31 = 2*2^4 - 1 is prime.

Links

Crossrefs

Cf. A007522, A127575-A127581, A127583-A127587.
A087522 is identical except for a(1).

Programs

  • Maple
    p:= 2: A[0]:= 2:
for n from 1 to 100 do
  if p+1 mod 2^n = 0 then A[n]:= p
  else
    p:=p+2^(n-1);
    while not isprime(p) do p:= p+2^n od:
    A[n]:= p;
  fi
od:
seq(A[i],i=0..100); # Robert Israel, Jan 13 2017
  • Mathematica
    a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 0, 50}]; a (* Artur Jasinski, Jan 19 2007 *)

Formula

a(n) << 37^n by Xylouris's improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Extensions

Edited by Don Reble, Jun 11 2007
Further edited by N. J. A. Sloane, Jul 03 2008
Showing 1-10 of 12 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.