A057735 -id:A057735 - 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

A102903 Primes of the form 3^k + 4.

Original entry on oeis.org

5, 7, 13, 31, 733, 19687, 59053, 31381059613, 205891132094653, 109418989131512359213, 1570042899082081611640534567, 323257909929174534292273980721360271853391

Offset: 1

Roger L. Bagula, Mar 01 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..17

Cf. A000040, A058958 (associated k).

Cf. Primes of the form 3^k + d: A057735 (d=2), this sequence (d=4), A102870 (d=8), A102907 (d=10), A102874 (d=14), A243437 (d=16), A102904 (d=20), A243438 (d=22), A243439 (d=26), A102906 (d=28).

[a: n in [0..100] | IsPrime(a) where a is 3^n+4]; // Vincenzo Librandi, Jul 19 2012

Select[Table[3^n+4,{n,0,200}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

a(n) = 3^A058958(n) + 4. - Elmo R. Oliveira, Nov 09 2023

Edited by Zak Seidov, Aug 29 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

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

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..16
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).

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).

[a: n in [0..300] | IsPrime(a) where a is 9^n+2];

Select[Table[9^n + 2, {n, 0, 300}], PrimeQ]

A144231 Prime numbers of the form 3^k +- 2 for k >= 1.

Original entry on oeis.org

5, 7, 11, 29, 79, 83, 241, 727, 6563, 19681, 59051, 4782971, 14348909, 31381059607, 282429536483, 2541865828331, 150094635296999123, 450283905890997361, 36472996377170786401, 1144561273430837494885949696429

Offset: 1

Reikku Kulon, Sep 15 2008

nonn

a(49) = 3^2224 - 2 and a(50) = 3^2521 - 2 are too big for a b-file. - Robert Israel, Nov 22 2015

Robert Israel, Table of n, a(n) for n = 1..48

Cf. A000040, A000668, A014232, A057735.

A:= NULL:
for k from 1 to 1000 do
  t:= 3^k;
  if isprime(t-2) then A:= A, t-2 fi;
  if isprime(t+2) then A:= A, t+2 fi;
od:
A; # Robert Israel, Nov 22 2015

Reap[For[k = 1, k <= 100, k++, p = 3^k-2; If[PrimeQ[p], Sow[p]]; If[PrimeQ[p+4], Sow[p+4]]]][[2, 1]] (* Jean-François Alcover, Dec 18 2013 *)

A182331 Primes of the form 6^k + 1.

Original entry on oeis.org

2, 7, 37, 1297

Offset: 1

Alex Ratushnyak, Apr 25 2012

nonn

No other terms for k < 3000. - T. D. Noe, Apr 25 2012

Apart from the first term, the exponents must be powers of two. Like Fermat primes, there are probably only finitely many terms. No more terms for k < 2^28 = 268435456. - Charles R Greathouse IV, Jul 16 2012

Wilfrid Keller, Prime factors of generalized Fermat numbers F_m(6) and complete factoring status

Cf. A057735, A182330.

Select[Table[6^n + 1, {n, 0, 100}], PrimeQ] (* T. D. Noe, Apr 25 2012 *)

ABC2 6^(2^$a)+1
a: from 0 to 18
// Charles R Greathouse IV, Jul 16 2012

A228026 Primes of the form 4^k + 3.

Original entry on oeis.org

7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, 19342813113834066795298819

Offset: 1

Vincenzo Librandi, Aug 11 2013

			67 is a term because 4^3 + 3 = 67 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..16

Cf. A089437 (associated k).

Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), this sequence (r=4, h=3), A228027 (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  4^n+3];

Select[Table[4^n + 3, {n, 0, 200}], PrimeQ]

a(n) = 4^A089437(n) + 3. - Elmo R. Oliveira, Nov 14 2023

Cross-references corrected by Robert Price, Aug 01 2017

A228032 Primes of the form 8^n + 3.

Original entry on oeis.org

11, 67, 4099, 32771, 262147, 1073741827, 19342813113834066795298819

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..14

Cf. A217354 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=8, h=3), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..300] | IsPrime(a) where a is  8^n+3];

Select[Table[8^n + 3, {n, 0, 300}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A234503 Number of ways to write n = k + m with k > 0 and m > 0 such that 3^(phi(k)/2 + phi(m)/12) + 2 is prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 4, 4, 2, 3, 2, 1, 3, 4, 8, 3, 4, 4, 4, 6, 3, 4, 6, 3, 5, 5, 3, 2, 2, 6, 5, 3, 2, 3, 7, 4, 3, 4, 4, 3, 4, 4, 4, 5, 2, 5, 2, 6, 5, 7, 3, 5, 7, 6, 13, 5, 7, 7, 10, 6, 8, 8, 9, 6, 7, 8, 6, 6, 5, 7, 9, 6, 7, 8, 10

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

It might seem that a(n) > 0 for all n > 14, but a(43905) = 0. If a(n) > 0 infinitely often, then there are infinitely many primes of the form 3^m + 2.
Similarly, it might seem that for n > 26 there is a positive integer k < n such that m = phi(k)/2 + phi(n-k)/12 is an integer with 3^m - 2 prime, but n = 41213 is a counterexample.
See also A234451 and A236358 for similar sequences.

			a(15) = 1 since 15 = 1 + 14 with 3^(phi(1)/2 + phi(14)/12) + 2 = 3 + 2 = 5 prime.
a(23) = 1 since 23 = 10 + 13 with 3^(phi(10)/2 + phi(13)/12) + 2 = 3^3 + 2 = 29 prime.
a(24) = 1 since 24 = 3 + 21 with 3^(phi(3)/2 + phi(21)/12) + 2 = 3^2 + 2 = 11 prime.
a(37) = 1 since 37 = 9 + 28 with 3^(phi(9)/2 + phi(28)/12) + 2 = 3^4 + 2 = 83 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Cf. A000010, A000040, A000244, A014232, A057735, A079363, A111974, A234344, A234346, A234347, A234361, A234451, A234470, A234475, A234504, A236358.

f[n_,k_]:=3^(EulerPhi[k]/2+EulerPhi[n-k]/12)+2
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A228029 Primes of the form 5^n + 6.

Original entry on oeis.org

7, 11, 31, 131, 631, 1220703131

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..15

Cf. A089142 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), this sequence (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  5^n+6];

Select[Table[5^n + 6, {n, 0, 200}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

cp's OEIS Frontend

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

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A102903 Primes of the form 3^k + 4.

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A182330 Primes of the form 5^k + 2.

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A228034 Primes of the form 9^n + 2.

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A144231 Prime numbers of the form 3^k +- 2 for k >= 1.

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Maple

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A182331 Primes of the form 6^k + 1.

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PFGW

A228026 Primes of the form 4^k + 3.

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Magma

Mathematica

Formula

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A228032 Primes of the form 8^n + 3.

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