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 1-5 of 5 results.

A002253 Numbers k such that 3*2^k + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From Zak Seidov, Mar 08 2009: (Start)
List is complete up to 3941000 according to the list of RB & WK.
So far there are only 4 primes: 2, 5, 41, 353. (End)

References

  • D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.
  • H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002254, A004119, A181565.
See A039687 for the actual primes.

Programs

  • PARI
    is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017
  • PARI
    A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<M. F. Hasler, Mar 03 2023

Formula

a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023

Extensions

Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 2009
Edited by N. J. A. Sloane, Mar 13 2009
a(47) and a(48) from the Ballinger & Keller web page, Joerg Arndt, Apr 07 2013
a(49) from https://t5k.org/primes/page.php?id=116922, Fabrice Le Foulher, Mar 09 2014
Terms moved from Data to b-file (Links), and additional term appended to b-file, by Jeppe Stig Nielsen, Oct 30 2020

A002254 Numbers k such that 5*2^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 4687, 5947, 13165, 23473, 26607, 125413, 209787, 240937, 819739, 1282755, 1320487, 1777515
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A050526.

Programs

Extensions

Corrected (removed incorrect term 40937) and added more terms (from http://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html), Joerg Arndt, Apr 07 2013

A238797 Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.

Original entry on oeis.org

0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0
Offset: 0

Views

Author

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 05 2014

Keywords

Comments

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:
For k = 0: 2, 3, 5, 7, 13, 17, ... Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ... Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3: Intersection of A059609 and A001771
for k = 4: 21, ... Intersection of A059610 and A002236
for k = 5: Intersection of A096817 and A001772
for k = 6: Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ... Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ... Intersection of A059611 and A001774
for k = 9: 5, 21, ... Intersection of A096819 and A001775
for k = 10: 7, 13, ... Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...

Examples

			a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.

Links

Crossrefs

Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).

Programs

  • Mathematica
    a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)

Extensions

a(0), a(19), a(20) corrected by Giovanni Resta, Mar 13 2014

A178835 Primes of the form 19 * 2^n - 1.

Original entry on oeis.org

37, 151, 607, 39845887, 41781441855487, 10696049115004927, 11760430373211112611541680127, 206891679087930585785731761318515072565247, 52964269846510229961147330897539858576703487, 888592995497188974267840378291500075950806986129407
Offset: 1

Views

Author

Kevin Batista (kevin762401(AT)yahoo.com), Jun 16 2010

Keywords

Comments

Main entry for this sequence is A001775.
Also primes of the form 38 * 4^n - 1, since even-numbered exponents yield numbers divisible by three.

Links

Programs

  • Magma
    [a: n in [0..250] | IsPrime(a) where a is (19 * 2^n - 1)]; // Vincenzo Librandi, Mar 08 2014
  • Mathematica
    Select[19*2^Range[140]-1,PrimeQ] (* Harvey P. Dale, Mar 07 2014 *)

Extensions

Edited by Charles R Greathouse IV, Aug 29 2010
Edited by Charles R Greathouse IV, Sep 02 2010
More terms from Harvey P. Dale, Mar 07 2014

A238749 Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.

Original entry on oeis.org

4, 8, 10, 12, 18, 32
Offset: 1

Views

Author

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

Keywords

Comments

a(7) > 350028.
Intersection of A059608 and A001770.
Exponents of Mersenne prime pairs {2^n - (2k + 1), (2k + 1)*2^n - 1}:
for k = 0: 2, 3, 5, 7, 13, 17, ... Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ... Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3: Intersection of A059609 and A001771
for k = 4: 21, ... Intersection of A059610 and A002236
for k = 5: Intersection of A096817 and A001772
for k = 6: Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ... Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ... Intersection of A059611 and A001774
for k = 9: 5, 21, ... Intersection of A096819 and A001775
for k = 10: 7, 13, ... Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...
for k = 14:

Examples

			a(1) = 4 because 2^4 - 5 = 11 and 5*2^4 - 1 = 79 are both primes.

Crossrefs

Cf. A000043, A001770, A059608, A237422, A238694, A238251, A238751, A238797.

Programs

  • Magma
    [n: n in [0..100] | IsPrime(2^n-5) and IsPrime(5*2^n-1)]; // Vincenzo Librandi, May 17 2015
  • Mathematica
    fQ[n_] := PrimeQ[2^n - 5] && PrimeQ[5*2^n - 1]; k = 1; While[ k < 15001, If[fQ@ k, Print@ k]; k++] (* Robert G. Wilson v, Mar 05 2014 *)
Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5 2^# - 1] &] (* Vincenzo Librandi, May 17 2015 *)
  • PARI
    isok(n) = isprime(2^n - 5) && isprime(5*2^n - 1); \\ Michel Marcus, Mar 04 2014
Showing 1-5 of 5 results.

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

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