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 62 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

A058959 Numbers k such that 3^k - 4 is prime.

Original entry on oeis.org

2, 3, 5, 21, 31, 37, 41, 53, 73, 101, 175, 203, 225, 455, 557, 651, 1333, 4823, 20367, 32555, 52057, 79371, 267267, 312155
Offset: 1

Views

Author

Robert G. Wilson v, Jan 13 2001

Keywords

Comments

From M. F. Hasler and Farideh Firoozbakht, Oct 30 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n)-4))=1 then m=3^(a(n)-1) (3^a(n)-4)Q is a solution of the equation sigma(x)=3(x+Q). This is a result of the following theorem.
Theorem : If for a prime q, Q is a (q-1)-perfect number and p=q^k-q-1 is a prime such that gcd(Q, p*q)=1, then m=p*q^(k-1)*Q is a solution of the equation sigma(x)=q(x+Q). The proof is easy. (End)
From M. F. Hasler and Farideh Firoozbakht, Dec 07 2009: (Start)
2 is the only even term of this sequence because if n is an even number greater than 2 then 3^n-4=(3^(n/2)-2)*(3^(n/2)+2) is composite.
We have also found the following generalization of this theorem. See comment lines of the sequence A171271.
Theorem : If for a prime q, Q is a (q-1)-perfect number and for some integers k and m, p=q^k-m*q-1 is a prime such that gcd(Q, p*q)=1, then x=p*q^(k-1)*Q is a solution of the equation sigma(x)=q(x+m*Q). The proof is easy. (End)

Links

Crossrefs

Cf. A014224, A171271. - M. F. Hasler and Farideh Firoozbakht, Dec 07 2009

Programs

  • Magma
    [n: n in [1..10^3]| IsPrime(3^n-4)]; // Vincenzo Librandi, Mar 24 2015
  • Mathematica
    Do[ If[ PrimeQ[3^n - 4], Print[n] ], {n, 1, 3000} ]
  • PARI
    for(n=1,10^3,if(ispseudoprime(3^n-4),print1(n,", "))) \\ Derek Orr, Mar 06 2015

Extensions

a(18)=4823, corresponding to a certified prime, from Ryan Propper, Jun 30 2005
a(19)=20367 from Ray Chandler, Jul 25 2011
a(20)=32555, a(21)=52057 from Henri Lifchitz, Jan 2005
a(22)=79371 from Ray Chandler, Jul 25 2011
a(23)=267267 from Roman Ilyukhin, Oct 17 2014
a(24)=312155 from Roman Ilyukhin, Feb 28 2015

A217137 Numbers k such that 3^k + 10 is prime.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 18, 36, 98, 114, 134, 138, 212, 252, 516, 1166, 2321, 2442, 2732, 4569, 8622, 8709, 16487, 22722, 25242, 29928, 32034, 33783, 34001, 44934, 50868, 77861, 90188, 102102, 171843, 178226, 273521
Offset: 1

Views

Author

Vincenzo Librandi, Oct 01 2012

Keywords

Comments

a(37) > 2*10^5. - Robert Price, Oct 23 2013
a(38) > 3*10^5. - Tyler NeSmith, Jan 16 2021

Crossrefs

Cf. A014224, A051783, A058958, A058959, A102907.

Programs

  • Mathematica
    Select[Range[0, 5000], PrimeQ[3^# + 10] &]
  • PARI
    for(n=0, 5*10^3, if(isprime(3^n+10), print1(n", ")))

Extensions

a(21)-a(36) from Robert Price, Oct 23 2013
a(37) from Tyler NeSmith, Jan 16 2021

A217347 Numbers k such that 3^k - 10 is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 17, 18, 21, 22, 36, 38, 41, 54, 56, 81, 92, 100, 106, 160, 310, 406, 560, 902, 5549, 9926, 12334, 19374, 19995, 20166, 39609, 62900, 186903, 244461
Offset: 1

Views

Author

Vincenzo Librandi, Oct 01 2012

Keywords

Comments

a(33) > 2*10^5. - Robert Price, Sep 07 2013
a(34) > 3*10^5. - Tyler NeSmith, Oct 03 2022

Crossrefs

Cf. A014224, A051783, A058958, A058959.

Programs

  • Magma
    /* The code produces the sequence up to 560: */ [n: n in [2..800] | IsPrime(3^n - 10)];
  • Mathematica
    Select[Range[2, 5000], PrimeQ[3^# - 10] &]
  • PARI
    for(n=2, 5*10^3, if(isprime(3^n-10), print1(n", ")))

Extensions

a(24)-a(32) from Robert Price, Sep 07 2013
a(33) from Tyler NeSmith, Oct 03 2022

A205647 Numbers k such that 3^k + 16 is prime.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 9, 12, 13, 15, 27, 31, 49, 57, 60, 75, 139, 147, 283, 327, 488, 604, 700, 825, 908, 1051, 1064, 1215, 5319, 9669, 10136, 16675, 25656, 28933, 35864, 47671, 68028, 73380, 186223, 194965, 221649, 233059, 240644, 513007, 543128, 551491, 648872, 989124, 994536
Offset: 1

Views

Author

Jonathan Vos Post, Jan 30 2012

Keywords

Comments

Indices of primes in A205646.
a(50) > 10^6. - Robert Price, Oct 28 2020

Examples

			57 is in the sequence because (3^57) + 16 = 1570042899082081611640534579 is prime.

Crossrefs

Cf. Sequences of numbers k such that 3^k + m is prime:
(m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
(m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
(m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
(m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
(m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
(m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Programs

Formula

{n: 3^n + 16 is in A000040} = {n: 3^n + 16 is prime} = {n: A000244(n) is prime} = {n: A205646(n) is prime}.

Extensions

5319 from Nicolas M. Perrault, Nov 10 2012
a(30)-a(40) from Robert Price, Oct 23 2013
a(41) discovered by Lelio R Paula, Nov 2016
a(42)-a(43) from Robert Price, Apr 02 2020
a(44)-a(46) from Robert Price, May 14 2020
a(47)-a(49) from Robert Price, Oct 28 2020

A217136 Numbers k such that 3^k + 8 is prime.

Original entry on oeis.org

1, 2, 4, 5, 8, 13, 14, 20, 38, 44, 77, 88, 124, 152, 244, 557, 2429, 4382, 6268, 18488, 75097, 81998, 96460, 143497
Offset: 1

Views

Author

Vincenzo Librandi, Oct 01 2012

Keywords

Comments

a(25) > 2*10^5. - Robert Price, Sep 25 2013

Crossrefs

Cf. A014224, A051783, A058958, A058959.

Programs

  • Mathematica
    Select[Range[0, 5000], PrimeQ[3^# + 8] &]
  • PARI
    for(n=1, 5*10^3, if(isprime(3^n+8), print1(n", ")))

Extensions

a(19)-a(24) from Robert Price, Sep 25 2013

A217135 Numbers k such that 3^k - 8 is prime.

Original entry on oeis.org

3, 4, 7, 8, 14, 20, 22, 62, 139, 254, 272, 430, 907, 1906, 2278, 2827, 3598, 6812, 15266, 20915, 26180, 26342, 27022, 48275, 65186, 69247, 86647
Offset: 1

Views

Author

Vincenzo Librandi, Oct 01 2012

Keywords

Comments

a(28) > 2*10^5. - Robert Price, Sep 02 2013

Crossrefs

Cf. A014224, A051783, A058958, A058959.

Programs

  • Mathematica
    Select[Range[2, 5000], PrimeQ[3^# - 8] &]
  • PARI
    for(n=2, 5*10^3, if(isprime(3^n-8), print1(n", ")))

Extensions

a(18)-a(27) from Robert Price, Sep 02 2013

A219035 Numbers k such that 3^k + 14 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 25, 98, 122, 153, 190, 258, 511, 549, 1703, 1790, 1870, 2418, 5226, 5258, 5626, 8550, 13174, 16718, 23669, 25215, 33447, 182566, 188286
Offset: 1

Views

Author

Nicolas M. Perrault, Nov 10 2012

Keywords

Comments

a(31) > 2*10^5. - Robert Price, Sep 27 2013

Crossrefs

Cf. Sequences of numbers k such that 3^k + m is prime:
(m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
(m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
(m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
(m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
(m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
(m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Programs

  • Mathematica
    Do[If[PrimeQ[3^n + 14], Print[n]], {n, 0, 10000}]
  • PARI
    is(n)=isprime(3^n+14) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(24)-a(30) from Robert Price, Sep 27 2013

A219038 Numbers k such that 3^k - 14 is prime.

Original entry on oeis.org

3, 4, 5, 8, 17, 19, 29, 124, 304, 640, 1205, 1549, 1805, 2492, 2945, 13075, 20237, 102763, 173755, 173828, 174040
Offset: 1

Views

Author

Nicolas M. Perrault, Nov 10 2012

Keywords

Comments

a(22) > 2*10^5. - Robert Price, Aug 31 2013

Crossrefs

Cf. Sequences of numbers k such that 3^k + m is prime:
(m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
(m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
(m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
(m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
(m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
(m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Programs

  • Mathematica
    Do[If[PrimeQ[3^n - 14], Print[n]], {n, 3, 3000}]
Select[Range[1000], PrimeQ[3^# - 14] &] (* Alonso del Arte, Nov 10 2012 *)
  • PARI
    is(n)=isprime(3^n-14) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(16)-a(21) from Robert Price, Aug 31 2013

A219040 Numbers k such that 3^k + 20 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 44, 55, 68, 71, 80, 123, 158, 213, 220, 272, 668, 725, 885, 1132, 1677, 2056, 2618, 3130, 3986, 6027, 8660, 11582, 12278, 14054, 62956, 103431, 120434, 123890, 181407
Offset: 1

Views

Author

Nicolas M. Perrault, Nov 10 2012

Keywords

Comments

a(40) > 2*10^5. - Robert Price, Oct 20 2013

Examples

			3^3 + 20 = 47 (prime), so 3 is in the sequence.

Crossrefs

Cf. Sequences of numbers k such that 3^k + m is prime:
(m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
(m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
(m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
(m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
(m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
(m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Programs

  • Mathematica
    Do[If[PrimeQ[3^n + 20], Print[n]], {n, 10000}]
  • PARI
    is(n)=isprime(3^n+20) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(32)-a(39) from Robert Price, Oct 20 2013
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