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 17 results. Next

A091516 Primes of the form 4^n - 2^(n+1) - 1.

Original entry on oeis.org

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087
Offset: 1

Views

Author

Eric W. Weisstein, Jan 17 2004

Keywords

Comments

Cletus Emmanuel calls these "Carol primes".
There are only 25 such primes below 4^1000. Terms beyond a(15) are too large to be displayed here: The sequence should be extended by listing the corresponding n-values in A091515. - M. F. Hasler, May 15 2008
Is there an explanation for the following observed pattern? Between groups of primes of roughly the same size, there is a gap of about the magnitude of these primes, i.e., the size roughly doubles (e.g., after the 16- and 17-digit primes, there is a 34-digit prime, then a 78-digit prime and some others up to 105 digits, then some 200- to 250-digit primes, then approximately 500 digits...). - M. F. Hasler, May 15 2008

Links

Crossrefs

Cf. A091515.

Programs

  • Mathematica
    lst={};Do[p=(2^n-1)^2-2;If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 160}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
  • PARI
    c=0;for(n=1,999,ispseudoprime(4^n-2^(n+1)-1)&write("b091516.txt",c++," ",4^n-2^(n+1)-1)) \\ M. F. Hasler, May 15 2008

Formula

a(k) = 4^A091515(k) - 2^(A091515(k) + 1) - 1 = (2^A091515(k) - 1)^2 - 2. - M. F. Hasler, May 15 2008

Extensions

Edited by Ray Chandler, Nov 15 2004

A098878 a(n) = (2^n - 1)^3 - 2.

Original entry on oeis.org

-2, -1, 25, 341, 3373, 29789, 250045, 2048381, 16581373, 133432829, 1070599165, 8577357821, 68669157373, 549554511869, 4397241253885, 35181150961661, 281462092005373, 2251748274470909, 18014192351838205
Offset: 0

Views

Author

Parthasarathy Nambi, Oct 13 2004

Keywords

Examples

			a(2) = (2^2 - 1)^3 - 2 = 25.

Links

Crossrefs

See A091515.

Programs

  • Mathematica
    Table[(2^n - 1)^3 - 2, {n, 0, 20}] (* Stefan Steinerberger, Feb 15 2006 *)
(2^Range[0,20]-1)^3-2 (* or *) LinearRecurrence[{15,-70,120,-64},{-2,-1,25,341},30] (* Harvey P. Dale, Jan 29 2016 *)
  • PARI
    a(n) = (2^n - 1)^3 - 2; \\ Joerg Arndt, Apr 30 2013

Formula

G.f.: (136*x^3-100*x^2+29*x-2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)). - Colin Barker, Apr 30 2013

Extensions

More terms from Stefan Steinerberger, Feb 15 2006

A269264 Numbers k such that (2^k-1)^2 - 2 is a semiprime.

Original entry on oeis.org

5, 9, 13, 16, 23, 24, 28, 35, 37, 38, 40, 47, 48, 51, 52, 57, 61, 65, 67, 70, 79, 83, 84, 85, 88, 90, 102, 111, 144, 148, 157, 162, 168, 169, 177, 181, 190, 237, 246, 298, 308, 346
Offset: 1

Views

Author

Vincenzo Librandi, Feb 21 2016

Keywords

Comments

a(43) >= 385. - Serge Batalov, Feb 26 2023

Examples

			a(1) = 5 because 31^2-2 = 959 = 7*137 which is semiprime.
a(2) = 9 because 511^2-2 = 261119 = 23*11353 which is semiprime.

Links

Crossrefs

Cf. A091515, A091516, A268574, A360993, A360994.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..120]| IsSemiprime(s) where s is (2^n-1)^2-2];
  • Mathematica
    Select[Range[120], PrimeOmega[(2^# - 1)^2 - 2] == 2 &]
  • PARI
    isok(n) = bigomega((2^n-1)^2-2) == 2; \\ Michel Marcus, Feb 22 2016

Extensions

a(29)-a(42) from Hugo Pfoertner, Aug 05 2019

A360993 Numbers k such that (2^k - 1)^3 + 2 is a semiprime.

Original entry on oeis.org

4, 5, 8, 12, 13, 18, 20, 29, 38, 56, 60, 62, 76, 82, 101, 118, 202, 210, 230, 276, 328, 332, 336, 338, 368
Offset: 1

Views

Author

Serge Batalov, Feb 27 2023

Keywords

Comments

a(26) >= 406.
438, 500, 526, 604, 648, 696 are also in this sequence, but their positions cannot be established before finding any factor for the values corresponding to the following "blockers": 406, 496, 528.
2382, 2733, 2910, 3368, 3508, 5338, 7705, 11185, 19905, 23814, 38545, 179294 are larger terms of this sequence, but their positions cannot be established. These produce "trivial" semiprimes where one prime is small (e.g., 3 or 11).

Examples

			a(1) = 4 because 15^3 + 2 = 3377 = 11 * 307, which is semiprime.
a(2) = 5 because 31^3 + 2 = 29793 = 3 * 9931, which is semiprime.

Links

Crossrefs

Cf. A091515, A091516, A100899, A100900, A268574, A269264, A360994.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..70]| IsSemiprime(s) where s is (2^n-1)^3+2];
  • Mathematica
    Select[Range[70], PrimeOmega[(2^# - 1)^3 - 2] == 2 &]
  • PARI
    isok(n) = bigomega((2^n-1)^3+2) == 2;

Extensions

a(20)-a(26) from Serge Batalov, Mar 03 2023

A364076 Numbers k such that (12^k - 1)^2 - 2 is prime.

Original entry on oeis.org

3, 29, 51, 7824, 15456, 22614, 28312, 47014, 68835
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 03 2023

Keywords

Comments

Such primes are sometimes called Carol primes of base 12.

Links

Crossrefs

Cf. A091515 (b=2), A100901 (b=6), A100903 (b=10), A100905 (b=14), A364078 (b=18), A364080 (b=20), A100907 (b=22).
Cf. A364077

Programs

  • Mathematica
    Select[Range[1500],PrimeQ[(12^#-1)^2-2]&] (* James C. McMahon, Jan 04 2024 *)
  • PARI
    for(k=1,1200,ispseudoprime((12^k-1)^2-2)&&print1(k,", "))

A364078 Numbers k such that (18^k - 1)^2 - 2 is prime.

Original entry on oeis.org

2, 8, 30, 98, 110, 185, 912, 2514, 4074, 10208, 15123, 19395, 69354
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 03 2023

Keywords

Comments

Such primes are sometimes called Carol primes of base 18.

Links

Crossrefs

Cf. A091515 (b=2), A100901 (b=6), A100903 (b=10), A364076 (b=12), A100905 (b=14), A364080 (b=20), A100907 (b=22).
Cf. A364079.

Programs

  • PARI
    for(k=1,1200,ispseudoprime((18^k-1)^2-2)&&print1(k,", "))

A364080 Numbers k such that (20^k - 1)^2 - 2 is prime.

Original entry on oeis.org

1, 2, 53, 183, 1281, 1300, 8041, 29936, 72820
Offset: 1

Views

Author

Jeppe Stig Nielsen, Jul 03 2023

Keywords

Comments

Such primes are sometimes called Carol primes of base 20.

Links

Crossrefs

Cf. A091515 (b=2), A100901 (b=6), A100903 (b=10), A364076 (b=12), A100905 (b=14), A364078 (b=18), A100907 (b=22).
Cf. A364081.

Programs

  • PARI
    for(k=1,1200,ispseudoprime((20^k-1)^2-2)&&print1(k,", "))

A260757 Least k > 0 such that M(n)^2 - 2k is prime, where M(n) = 2^n - 1 = A000225(n).

Original entry on oeis.org

1, 2, 1, 1, 1, 4, 1, 1, 7, 10, 1, 10, 1, 10, 5, 1, 14, 24, 1, 1, 13, 1, 16, 3, 82, 1, 19, 1, 23, 94, 64, 58, 7, 6, 14, 3, 46, 22, 5, 13, 107, 69, 38, 90, 59, 75, 104, 25, 4, 10, 14, 4, 44, 10, 5, 1, 77, 81, 85, 94, 71, 9, 14, 111, 13, 27, 20, 9, 37, 6, 5, 4, 62, 12, 38, 4, 37
Offset: 0

Views

Author

M. F. Hasler, Jul 30 2015

Keywords

Comments

For n = 0 and n = 1, no k > 0 can yield a positive prime, the given values are the smallest to yield the opposite of a positive prime: M(0)^2 - 2*1 = 0 - 2 = -2 and M(1)^2 - 2*2 = 1 - 4 = -3.

Examples

			For n = 2, M(2) = 2^2 - 1 = 3 and 3*3 - 2k = 7 is a prime for k=1, thus a(2) = 1.
For n = 3, M(3) = 2^3 - 1 = 7 and 7*7 - 2k = 47 is a prime for k=1, thus a(3) = 1.
For n = 4, M(4) = 2^4 - 1 = 15 and 15*15 - 2k = 223 is a prime for k=1, thus a(4) = 1.
For n = 5, M(5) = 2^5 - 1 = 31 and 31*31 - 2k = 953 is prime for k=4 and no smaller k, thus a(5) = 4.

Links

Crossrefs

Cf. A091515 (a(n)=1 for n > 0), A260758.

Programs

  • Maple
    f:= proc(n) local r;
  r:= (2^n-1)^2;
  (r - prevprime(r))/2
end proc:
f(0):=1: f(1):= 2:
map(f, [$0..100]); # Robert Israel, Apr 02 2020
  • Mathematica
    f[n_] := Module[{r = (2^n - 1)^2}, (r - NextPrime[r, -1])/2 ];
f[0] = 1; f[1] = 2;
f /@ Range[0, 100] (* Jean-François Alcover, Jul 28 2020, after Robert Israel *)
  • PARI
    a(n)={n>1&&for(k=1,9e9,ispseudoprime((2^n-1)^2-2*k)&&return(k));n+1}

Formula

a(n) = 1 for n=0 or n in A091515.

A098879 a(n) = (2^n - 1)^5 - 2.

Original entry on oeis.org

-2, -1, 241, 16805, 759373, 28629149, 992436541, 33038369405, 1078203909373, 34842114263549, 1120413075641341, 35940921946155005, 1151514816750309373, 36870975646169341949, 1180231376725002502141, 37773167607267111108605, 1208833588708967444709373
Offset: 0

Views

Author

Parthasarathy Nambi, Oct 13 2004

Keywords

Comments

5th-power analog of what for exponent 2 is A093112 (2^n-1)^2 - 2 = 4^n - 2^{n+1} - 1 and exponent 3 is A098878 (2^n - 1)^3 - 2. Primes include a(n) for n = 0, 2, 5, 6. These are "near-5th-power prime." Semiprimes include a(n) for n = 3, 8, 9, 10, 13, 15, 21, 29, 33, 40. - Jonathan Vos Post, May 03 2006

Examples

			If n=2, (2^2 - 1)^5 - 2 = 241 (a prime).

Links

Crossrefs

Cf. A091516, A091515, A098878, A091514.

Programs

  • Mathematica
    (2^Range[0,20]-1)^5-2 (* or *) LinearRecurrence[{63,-1302,11160,-41664,64512,-32768},{-2,-1,241,16805,759373,28629149},20] (* Harvey P. Dale, Nov 03 2016 *)
  • PARI
    a(n)=(2^n-1)^5-2 \\ Charles R Greathouse IV, Feb 19 2016

Formula

G.f.: (-2+125*x-2300*x^2+22640*x^3-57728*x^4+66560*x^5)/((-1+x)(-1+32*x)(-1+16*x)(-1+8*x)(-1+4*x)(-1+2*x)). - R. J. Mathar, Nov 14 2007

Extensions

More terms from Jonathan Vos Post, May 03 2006
Edited by N. J. A. Sloane, Sep 30 2007

A117921 Primes of the form (2^k - 1)^3 - 2.

Original entry on oeis.org

3373, 29789, 133432829, 8577357821, 281462092005373
Offset: 1

Views

Author

Jonathan Vos Post, May 03 2006

Keywords

Comments

Exponent-3 analog of what for exponent 2 is A091516 Carol primes (2^n-1)^2 - 2 = 4^n - 2^{n+1} - 1. Hence this is a type of "near-cube primes."

Examples

			a(1) = (2^4 - 1)^3 - 2 = 3373 is prime.
a(2) = (2^5 - 1)^3 - 2 = 29789 is prime.
a(3) = (2^9 - 1)^3 - 2 = 133432829 is prime.
a(4) = (2^11 - 1)^3 - 2 = 8577357821 is prime.

Links

Crossrefs

Cf. A091516, A091515, A098878, A091514.

Programs

  • Mathematica
    Select[(2^Range[20]-1)^3-2,PrimeQ] (* Harvey P. Dale, Oct 22 2016 *)

Formula

A098878 INTERSECTION A000040. {(2^k - 1)^3 - 2 iff prime}.
Showing 1-10 of 17 results. Next

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

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