A087885 -id:A087885 - cp's OEIS frontend

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

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

Jud McCranie, Dec 09 1999

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]

			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.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri and Renaud Lifchitz, PRP Records.

Cf. A057735, A087885, A014224, A058959.

A051783 = Select[Range[0, 20000], PrimeQ[3^# + 2] &]

is(n)=ispseudoprime(3^n+2) \\ Charles R Greathouse IV, Mar 21 2013

{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

A090649 Numbers k such that 9^k + 2 is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 12, 13, 18, 49, 55, 63, 193, 247, 610, 929, 2173, 3479, 5494, 11158, 16754, 30920, 47752, 50702, 53725, 68122, 89214, 180804

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Feb 02 2004

All terms are the exact halves of the even terms in A051783. - Alexander Adamchuk, Mar 02 2008

a(29) > 2*10^5. - Robert Price, Aug 18 2014

			9^13 + 2 = 2541865828331 is prime, so 13 is a term.

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A051783 (3^k + 2 is prime), A087885 (5^k + 2 is prime).

Do[ If[ PrimeQ[9^n + 2], Print[n]], {n, 1, 2250}] (* Robert G. Wilson v, Feb 06 2004 *)

for(i=0,700,if(isprime(9^i+2),print(i)))

More terms from mohammed bouayoun (bouyao(AT)wanadoo.fr), Feb 04 2004
Further terms from Robert G. Wilson v and Ray Chandler, Feb 06 2004
a(20) - a(22) found by Henri Lifchitz, a(23) found by Wojciech Florek. - Jason Earls, Feb 25 2008
a(24)-a(27) from A051783 by Ray Chandler, Aug 06 2011
a(28) from Robert Price, Aug 18 2014
Removal of erroneous term (97715) by Robert Price, Aug 19 2014

A182330 Primes of the form 5^k + 2.

Original entry on oeis.org

3, 7, 127, 762939453127

Offset: 1

Alex Ratushnyak, Apr 25 2012

nonn

Next term has 100 digits.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A057735, A087885, A242328.

Select[Table[5^n + 2, {n, 0, 500}], PrimeQ] (* T. D. Noe, Apr 25 2012 *)

a(n) = A242328(A087885(n)). - Amiram Eldar, Jul 23 2025

A087886 Numbers n such that 29^n + 2 is prime.

Original entry on oeis.org

0, 1, 3, 63, 87, 189, 239, 605, 2099, 3667, 5029, 20025, 45285, 99167

Offset: 1

Donald S. McDonald, Oct 13 2003

After 0, all terms are odd. [Bruno Berselli, Oct 02 2014]

			a(1) = 1 is a member because 29^1 + 2 = 31 is prime.
29^3+2 = 24391 is prime, so 3 is in the sequence.

Henri & Renaud Lifchitz, PRP Records.

Cf. A051783, n such that 3^n +2 is prime.

Cf. A087885, 5^n +2 is prime.

for( n = 0,239, if( isprime( 29^n +2), print( n), ))

a(8) from Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 07 2004
a(9) - a(11), corresponding to probable primes, from Ryan Propper, Jul 03 2005
Edited by T. D. Noe, Oct 30 2008
a(12) from Ray Chandler, Aug 01 2011
a(13) from Ray Chandler, Aug 02 2011
a(14) from Ray Chandler, Aug 08 2011

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

Vincenzo Librandi, Sep 28 2014

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

Henri & Renaud Lifchitz, PRP Records, search for 33^n+2

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).

[n: n in [0..350]| IsPrime( 33^n + 2 )];

A247957:=n->`if`(isprime(33^n+2),n,NULL): seq(A247957(n), n=0..1000); # Wesley Ivan Hurt, Sep 28 2014

Select[Range[0,1000], PrimeQ[33^# + 2] &]

is(n)=ispseudoprime(33^n+2) \\ Charles R Greathouse IV, Feb 17 2017

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

Alexander Adamchuk, Mar 02 2008

No further terms < 100000. - Ray Chandler, Aug 05 2011

a(23) > 2*10^5. - Robert Price, Jun 23 2015

Henri & Renaud Lifchitz, PRP Records.

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.

Do[ f = 15^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 371} ]

is(n)=ispseudoprime(15^n+2) \\ Charles R Greathouse IV, Feb 17 2017

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

Alexander Adamchuk, Mar 02 2008

No further terms < 100000. - Ray Chandler, Aug 11 2011

a(15) > 2*10^5. - Robert Price, Jul 14 2015

Henri & Renaud Lifchitz, PRP Records.

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.

Do[ f = 21^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 310} ]

is(n)=ispseudoprime(21^n+2) \\ Charles R Greathouse IV, Feb 17 2017

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

Alexander Adamchuk, Mar 02 2008

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

Henri & Renaud Lifchitz, PRP Records.

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.

Do[ f = 23^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 323} ]

is(n)=ispseudoprime(23^n+2) \\ Charles R Greathouse IV, Feb 17 2017

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

Alexander Adamchuk, Mar 02 2008

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

Henri & Renaud Lifchitz, PRP Records.

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

Do[ f = 27^n + 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n, 1, 2500} ]

is(n)=ispseudoprime(27^n+2) \\ Charles R Greathouse IV, Feb 17 2017

a(15)-a(17) from A051783 by Ray Chandler, Aug 06 2011

a(18) from Robert Price, May 12 2019

A217133 Numbers n such that 5^n + 8 is prime.

Original entry on oeis.org

1, 95, 335, 3155, 28651, 91135

Offset: 1

Vincenzo Librandi, Oct 01 2012

Naturally these numbers are odd since (6-1)^(2n)+8 is divisible by 3. - Bruno Berselli, Oct 04 2012
a(7) > 10^5. - Robert Price, Feb 03 2014
a(7) > 5*10^5. - Tyler NeSmith, Apr 24 2022

Cf. A087885, A089142, A124621.

Select[Range[1, 15000, 2], PrimeQ[5^# + 8] &]

for(n=1, 5*10^3, if(isprime(5^n+8), print1(n", ")))

a(5)-a(6) from Robert Price, Feb 03 2014

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

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A090649 Numbers k such that 9^k + 2 is prime.

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A182330 Primes of the form 5^k + 2.

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A087886 Numbers n such that 29^n + 2 is prime.

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

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

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

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