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

A051783 Numbers k such that 3^k + 2 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494, 1131, 1220, 1503, 1858, 4346, 6958, 7203, 10988, 22316, 33508, 43791, 45535, 61840, 95504, 101404, 106143, 107450, 136244, 178428, 361608, 504206, 1753088
Offset: 1

Views

Author

Jud McCranie, Dec 09 1999

Keywords

Comments

From Farideh Firoozbakht and M. F. Hasler, Dec 06 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n) + 2)) = 1, then x = 3^(a(n) - 1)*(3^a(n) + 2)*Q is a solution of the equation sigma(x) = 3(x - Q). This is a result of the following theorem:
Theorem: If Q is a (q-1)-perfect number for some prime q, then for all integers t, the equation sigma(x) = q*x - (2t+1)*Q has the solution x = q^(k-1)*p*Q whenever k is a positive integer such that p = q^k + 2t is prime, gcd(q^(k-1), p) = 1 and gcd(q^(k-1)*p,Q) = 1.
Note that by taking t = -1/2(m*q+1), this theorem gives us some solutions of the equation sigma(x) = q *(x + m*Q). See comment lines of the sequence A058959. (End)
No further terms < 200000. - Ray Chandler, Jul 31 2011
A090649 implies that 361608 is a member of this sequence. - Robert Price, Aug 18 2014
No further terms < 320000. - Luke W. Richards, Mar 04 2018
a(45) and a(46) are probable primes because a primality certificate has not yet been found. They have been verified PRP with mprime. - Luke W. Richards, May 04 2018
No further terms < 1300000. - Luke W. Richards, May 17 2018
No further terms < 1400000. - Luke W. Richards, Jul 28 2020
Conjecture: The number n = 3^k + 2 is prime if and only if 2^((n-1)/2) == -1 (mod n). - Maheswara Rao Valluri, Jun 01 2020. [Note that this is an if and only if assertion, so it does not follow from Fermat's Little Theorem. - N. J. A. Sloane, Sep 07 2020]

Examples

			3^8 + 2 = 6563 is prime, so 8 is in the sequence.
3^26 + 2 = 2541865828331, a prime number, so 26 is in the sequence.

Links

Crossrefs

Cf. A057735, A087885, A014224, A058959.

Programs

Extensions

{4346, 6958, 7203} from David J. Rusin, Sep 29 2000
10988 from Ray Chandler, Nov 21 2004
{22316, 33508} found by Henri Lifchitz, Sep-Oct 2002
{43791, 45535, 61840} found by Henri Lifchitz, Oct-Nov 2004
95504 found by Wojciech Florek Dec 15 2005. - Alexander Adamchuk, Mar 02 2008
Edited by N. J. A. Sloane, Dec 19 2009
{101404, 106143, 107450, 136244} from Mike Oakes, Nov 2009
178428 from Ray Chandler, Jul 29 2011
a(45)-a(46) from Luke W. Richards, May 04 2018
a(47) from Paul Bourdelais, Mar 29 2022

A228034 Primes of the form 9^n + 2.

Original entry on oeis.org

3, 11, 83, 6563, 59051, 4782971, 282429536483, 2541865828331, 150094635296999123, 57264168970223481226273458862846808078011946891, 30432527221704537086371993251530170531786747066637051
Offset: 1

Views

Author

Vincenzo Librandi, Aug 11 2013

Keywords

Links

Crossrefs

Cf. A004051 (primes of the form 2^a+3^b), A057735 (primes of the form 3^n+2), A090649 (associated n), A104070 (primes of the form 2^n+9), A159352 (primes of the form 10^n+3), A176495 (primes of the form 27^n+2), A182330 (primes of the form 5^n+2).

Programs

  • Magma
    [a: n in [0..300] | IsPrime(a) where a is 9^n+2];
  • Mathematica
    Select[Table[9^n + 2, {n, 0, 300}], PrimeQ]

A247957 Numbers k such that 33^k + 2 is prime.

Original entry on oeis.org

0, 2, 26, 60, 218, 248, 399, 1175, 1244, 2670, 9300, 45216, 144412
Offset: 1

Views

Author

Vincenzo Librandi, Sep 28 2014

Keywords

Comments

Some terms correspond to probable primes. Lifchitz link shows that Ray Chandler discovered 9300, and Lelio R Paula found that 45216 is in the sequence. - Jens Kruse Andersen, Sep 29 2014
Terms in the similar sequence, 53^k+2, begin with 0, 7, 14483 with the next term > 2*10^5. - Robert Price, Mar 28 2015
Confirmed that 45216 is a(12). - Robert Price, Apr 14 2015
a(14) > 2*10^5. - Robert Price, Apr 14 2015

Links

Crossrefs

Cf. numbers n such that k^n+2 is prime: A051783 (k=3), A087885 (k=5), A090649 (k=9), A109076 (k=11), A138048 (k=15), A113480 (k=17), A138049 (k=21), A138050 (k=23), A138051 (k=27), A087886 (k=29), this sequence (k=33), A247958 (k=35), A247959 (k=39), A247960 (k=41), A247961 (k=45); (0, 113) for k=47; A247962 (k=51); A247963 (k=57), A113481 (k=59).

Programs

  • Magma
    [n: n in [0..350]| IsPrime( 33^n + 2 )];
  • Maple
    A247957:=n->`if`(isprime(33^n+2),n,NULL): seq(A247957(n), n=0..1000); # Wesley Ivan Hurt, Sep 28 2014
  • Mathematica
    Select[Range[0,1000], PrimeQ[33^# + 2] &]
  • PARI
    is(n)=ispseudoprime(33^n+2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(8)-a(11) from Jens Kruse Andersen, Sep 29 2014
a(12)-a(13) from Robert Price, Apr 14 2015

A138048 Numbers k such that 15^k + 2 is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 11, 16, 20, 52, 75, 106, 112, 132, 371, 3264, 3424, 5477, 7516, 10365, 44557, 150706
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2008

Keywords

Comments

No further terms < 100000. - Ray Chandler, Aug 05 2011
a(23) > 2*10^5. - Robert Price, Jun 23 2015

Links

Crossrefs

Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138049, A138050, A138051, A087886, A113481.

Programs

  • Mathematica
    Do[ f = 15^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 371} ]
  • PARI
    is(n)=ispseudoprime(15^n+2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(16)-a(19) from Ray Chandler, Jul 30 2011
a(20) found by Lelio R Paula, Dec 2006
a(21) from Ray Chandler, Jul 31 2011
a(22) from Robert Price, Jun 23 2015

A138049 Numbers k such that 21^k + 2 is prime.

Original entry on oeis.org

0, 1, 2, 4, 7, 24, 40, 112, 310, 1026, 1286, 36566, 43717, 53753
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2008

Keywords

Comments

No further terms < 100000. - Ray Chandler, Aug 11 2011
a(15) > 2*10^5. - Robert Price, Jul 14 2015

Links

Crossrefs

Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138050, A138051, A087886, A113481.

Programs

  • Mathematica
    Do[ f = 21^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 310} ]
  • PARI
    is(n)=ispseudoprime(21^n+2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

1026 from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
1286 and 36566 from Ray Chandler, Jul 31 2011
43717 from Ray Chandler, Aug 01 2011
53753 from Ray Chandler, Aug 02 2011

A138050 Numbers k such that 23^k + 2 is prime.

Original entry on oeis.org

0, 11, 39, 323, 12415, 14655, 27679
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2008

Keywords

Comments

No further terms < 100000. - Ray Chandler, Aug 03 2011

Links

Crossrefs

Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138051, A087886, A113481.

Programs

  • Mathematica
    Do[ f = 23^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 323} ]
  • PARI
    is(n)=ispseudoprime(23^n+2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(5)-a(7) from Ray Chandler, Aug 01 2011

A138051 Numbers k such that 27^k + 2 is prime.

Original entry on oeis.org

0, 1, 5, 8, 12, 21, 41, 42, 81, 105, 121, 377, 501, 2401, 14597, 35381, 59476, 120536
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2008

Keywords

Comments

All terms are the exact thirds of terms of A051783 that are divisible by 3.

Links

Crossrefs

Cf. A051783 (3^k + 2 is prime), A087885 (5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138050, A087886, A113481.
Cf. A176495.

Programs

  • Mathematica
    Do[ f = 27^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 2500} ]
  • PARI
    is(n)=ispseudoprime(27^n+2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(15)-a(17) from A051783 by Ray Chandler, Aug 06 2011
a(18) from Robert Price, May 12 2019

A138066 Least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 11, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 113, 0, 1, 7, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 12, 1, 0, 1, 3, 0, 1, 255, 0, 8, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 2, 15, 0, 2, 1, 0, 1, 23, 0, 1, 1, 0, 4, 3, 0, 1, 1, 0, 3, 1, 0, 136, 1, 0, 1
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2008

Keywords

Comments

a(3n+1) = 0 for n > 0.
a(84) > 100000. - Ray Chandler, Aug 10 2011

Crossrefs

Cf. A084713 (smallest prime of the form (2n-1)^k + 2, or 0 if no such number exists).
Cf. A138067 (least k > 1 such that (2n-1)^k + 2 is prime, or 0 if no such number exists).
Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138050, A138051, A087886, A113481.

A217384 Numbers k such that 9^k + 4 is prime.

Original entry on oeis.org

0, 1, 3, 5, 11, 15, 21, 87, 99, 281, 497, 2919, 6849, 7365, 8483, 49317, 58611
Offset: 1

Views

Author

Vincenzo Librandi, Oct 04 2012

Keywords

Comments

Contribution from Bruno Berselli, Oct 04 2012: (Start)
Contains exactly the halved even terms of A058958.
Naturally, apart from the first term, these numbers are odd since (10-1)^(2n)+4 is divisible by 5. (End)
a(18) > 10^5. - Tyler NeSmith, May 05 2021

Crossrefs

Cf. A058958, A090649.

Programs

  • Mathematica
    Select[Range[0, 3000], PrimeQ[9^# + 4] &]
  • PARI
    is(n)=ispseudoprime(9^n+4) \\ Charles R Greathouse IV, Feb 20 2017

Extensions

a(13)-a(14) from Bruno Berselli, Oct 04 2012
a(15)-a(17) from Tyler NeSmith, May 05 2021

A138067 Least k > 1 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

2, 2, 3, 0, 2, 5, 0, 2, 105, 0, 2, 11, 0, 5, 3, 0, 2, 15, 0, 2, 9, 0, 2, 113, 0, 5, 7, 0, 2, 27, 0, 3, 3, 0, 3, 3, 0, 12, 61, 0, 2, 3, 0, 4, 255, 0, 8, 63, 0, 2, 9, 0, 2, 3473, 0, 2, 3, 0, 2, 15, 0, 2, 87, 0, 3, 23, 0, 36, 1861, 0, 4, 3, 0, 2, 5, 0, 3, 7, 0, 136, 425, 0, 11
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2008

Keywords

Comments

a(3n+1) = 0 for n > 0.
a(84) > 100000. - Ray Chandler, Aug 10 2011

Crossrefs

Cf. A084713 (smallest prime of the form (2n-1)^k + 2, or 0 if no such number exists).
Cf. A138066 (least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists).
Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138050, A138051, A087886, A113481.

Extensions

a(54)-a(83) from Donovan Johnson, Oct 29 2008
Showing 1-10 of 12 results. Next

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