A003306 -id:A003306 - cp's OEIS frontend

A003307 Numbers k such that 2*3^k - 1 is prime.

1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104

Offset: 1

N. J. A. Sloane

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
Steven Harvey, Williams Primes
W. Keller and J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A002235, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Cf. A079363 (primes of the form 2*3^k - 1), A003306 (k such that 2*3^k + 1 is prime).

for(n=1,1e4,if(ispseudoprime(2*3^n-1),print1(n", "))) \\ Charles R Greathouse IV, Jul 16 2011

More terms from Douglas Burke (dburke(AT)nevada.edu)
More terms from T. D. Noe, Aug 24 2005
Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(35) from Borys Jaworski, Sep 02 2011
a(36) from Borys Jaworski, Feb 13 2012
a(37) from Jeppe Stig Nielsen, Sep 28 2018

A079363 Primes of the form 2*3^k - 1.

Original entry on oeis.org

5, 17, 53, 4373, 13121, 1062881, 6973568801, 188286357653, 15251194969973, 100063090197999413, 1046695266054721074427023041, 763040848953891663257299797617, 556256778887387022514571552463521

Offset: 1

Cino Hilliard, Feb 15 2003

nonn

Sum of reciprocals = 0.2779972845973183835923785945..

Vincenzo Librandi, Table of n, a(n) for n = 1..20

Cf. A003306 (n such that 2*3^n+1 is prime), A003307 (n such that 2*3^n-1 is prime).

[a: n in [1..200] | IsPrime(a) where a is  2*3^n-1 ]; // Vincenzo Librandi, Dec 09 2011

select(isprime,[2*3^k-1$k=0..200]); # Muniru A Asiru, Nov 24 2018

Mathematica
```
Select[2*3^Range[100]-1, PrimeQ]
```

\\ Primes in the sequence of sums of alternating powers of 3
pseq3(n) = { j=a=1; p=1; sr=0; while(j<=n, a = a + 3^(p); if(isprime(a),print1(a", "); sr+=1.0/a; ); a = a+3^(p-1); if(isprime(a),print1(a", "); sr+=1.0/a; ); p+=1; j+=2; ); print(); print(sr); }

A111974 Primes of the form 2*3^k + 1.

Original entry on oeis.org

3, 7, 19, 163, 487, 1459, 39367, 86093443, 258280327, 411782264189299, 116299474006080119380780339, 3140085798164163223281069127, 84782316550432407028588866403, 20602102921755074907947094535687, 1910009901593650473786381403548828023839870277948686259673707683

Offset: 1

T. D. Noe, Aug 24 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..23
Hartosh Singh Bal and Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019-2020.

Cf. A003306 (k such that 2*3^k + 1 is prime), A003307 (k such that 2*3^k - 1 is prime), A052919.

Mathematica
```
Select[2*3^Range[100]+1, PrimeQ]
```

a(n) = A052919(A003306(n)+1). - Amiram Eldar, Jul 18 2025

a(15) from Amiram Eldar, Jul 18 2025

A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

Original entry on oeis.org

3, 5, 7, 17, 19, 43, 101, 157, 163, 257, 487, 1459, 2029, 4423, 6163, 14407, 19183, 22651, 23549, 26407, 37057, 39367, 62501, 65537, 77659, 113233, 121453, 143263, 208393, 292141, 342733, 375157, 412807, 527803, 564899, 590593, 697049, 843643

Offset: 1

T. D. Noe, Aug 15 2003

nonn

It is usually the case that, for prime p and k > 1, the first time the totient function phi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Totient Valence Function

Cf. A002383 (primes of the form n^2 + n + 1, which is the same as n^2 - n + 1).

Cf. A019434 (Fermat primes), A003306 (2*3^n + 1 is prime), A056799 (8*9^n + 1 is prime), A056797 (9*10^n + 1 is prime), A087139 (least k such that p^k - p^(k-1) + 1 is prime for p = prime(n)).

lst={}; maxNum=10^6; n=1; While[p=Prime[n]; p^2-p+1

A239676 Least k such that k*3^n+1 is prime.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 8, 6, 2, 8, 28, 10, 12, 4, 4, 2, 2, 10, 20, 26, 24, 8, 48, 16, 34, 14, 14, 18, 6, 2, 26, 26, 14, 22, 26, 16, 22, 12, 4, 62, 64, 68, 88, 70, 56, 34, 96, 32, 50, 20, 24, 8, 6, 2, 18, 6, 2, 8, 6, 2, 42, 14, 18, 6, 2, 98, 66, 22, 70, 74, 80, 68, 52

Offset: 0

Derek Orr, Mar 23 2014

nonn

All numbers in this sequence, except for a(0), are even.

			1*3^1+1 = 4 is not prime. 2*3^1+1 = 7 is prime. Thus, a(1) = 2.
1*3^3+1 = 28 is not prime. 2*3^3+1 = 57 is not prime. 3*3^3+1 = 82 is not prime. 4*3^3+1 = 109 is prime. Thus, a(3) = 4.

Marius A. Burtea, Table of n, a(n) for n = 0..1000

Cf. A003306 (where k=2), A035050 (k*2^n+1 is prime).

sol:=[];m:=1; for n in [0..73] do k:=0; while not IsPrime(k*3^n+1) do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Jun 05 2019

lk[n_]:=Module[{k=1,t=3^n},While[!PrimeQ[k*t+1],k++];k]; Array[lk,80,0] (* Harvey P. Dale, May 11 2025 *)

for(n=0, 100, k=0; while(!isprime(k*3^n+1), k++); print1(k, ", ")) \\ Colin Barker, Mar 24 2014

import sympy
from sympy import isprime
def Pow3(n):
  for k in range(10**4):
    if isprime(k*(3**n)+1):
      return n
n = 1
while n < 100:
  print(Pow3(n))
  n += 1

A305237 Numbers m such that m, m+1 and m+2 all have primitive roots.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 17, 25, 81, 241

Offset: 1

Jianing Song, Jun 04 2018

nonn

Start of run of 3 consecutive numbers in A033948.
The next term is 3^541 - 2, which is too large to be included here. No more terms below 3^100000, or approximately 1.33*10^47712.
There is a multiple of 4 in every four consecutive positive integers and it clearly has no primitive roots if it is larger than 4. Again, there is a multiple of 3 in every three consecutive positive integers, so it must be a power of 3 or two times a power of 3, and the other two numbers must be odd prime powers or two times odd prime powers.
According to Pillai's conjecture, there're only finitely many solutions to |3^a - p^b| = 2, |3^a - 2*p^b| = 1, |p^a - 2*3^b| = 1 with a,b >= 2, p odd primes (no solution other than 3^3 - 5^2 = 2, 3^5 - 2*11^2 = 1 below 3^100000). So beyond (25, 26, 27) and (241, 242, 243), it's very likely that all three consecutive numbers with primitive roots are of the form (3^i, 3^i + 1, 3^i + 2), (3^j - 2, 3^j - 1, 3^j), (2*3^k - 1, 2*3^k, 2*3^k + 1) such that (3^i + 1)/2, 3^i + 2, 3^j - 2, (3^j - 1)/2, 2*3^k - 1, 2*3^k + 1 are primes, which only produces one more solution (3^541 - 2, 3^541 - 1, 3^541) below 3^1000000.

			81, 82, 83 all have primitive roots (in fact, their least common primitive root is 47), so 81 is a term.
Note that A014224 and A028491 have a term 541 in common, so 3^541 - 2, 3^541 - 1 and 3^541 all have primitive roots, so 3^541 - 2 is a term.

Jinyuan Wang, Table of n, a(n) for n = 1..11

Cf. A003306, A003307, A014224, A028491, A051783, A171381, A319450.

A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists.

Original entry on oeis.org

0, 6, 153, 66, 0, 3916, 0, 1770, 2556, 327645, 0, 1540, 0, 893862621, 8199225, 17766, 0, 76636, 0, 12720, 662976, 2096128, 0, 10296, 3357936, 416798777159765703, 6221628, 3611328, 0, 1734453, 0, 303810, 111576864636, 1420010137134674578503, 18051523357140153

Offset: 1

Jon E. Schoenfield, Dec 05 2018

nonn

The only primes p for which a(p) > 0 are those for which both 2*3^(p-1) - 1 and 2*3^(p-1) + 1 are prime: 2, 3, and any other primes p such that p-1 appears both in A003307 and A003306. (If such a prime p > 3 exists, then p exceeds 1360105.)

Conjecture: The only primes p for which a(p) > 0 are 2 and 3.

			For n=1, the only triangular number with exactly 2*1 = 2 divisors is T(2) = 2*(2+1)/2 = 3 (the only triangular number that is prime); thus, exists no pair of consecutive triangular numbers having exactly 2 divisors, so a(1)=0.
a(2) is 6 because T(3) = 3*(3+1)/2 = 6 and T(4) = 4*(4+1)/2 = 10 are the first two consecutive triangular numbers having exactly 2*2 = 4 divisors.

Cf. A000005, A000217, A063440, A081978, A319035.

Cf. A003306, A003307.

A056802 Numbers k such that 2*9^k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 8, 15, 27, 30, 66, 90, 160, 348, 391, 411, 626, 727, 2740, 3921, 6048, 6891, 8860, 21978, 32411, 41390, 52553, 82948, 95907, 264840, 537363, 543056, 587616, 638931, 673271, 1561518

Offset: 1

Robert G. Wilson v, Aug 22 2000

Cf. A003306 (k such that 2*3^k + 1 is prime).

Do[ If[ PrimeQ[ 2*9^n + 1], Print[ n ]], {n, 0, 1000}]

is(n)=ispseudoprime(2*9^n+1) \\ Charles R Greathouse IV, Jun 12 2017

More terms from Rick L. Shepherd, Apr 22 2003
a(21)-a(31) from David Broadhurst, Feb 22 2010
a(32)-a(34) from Paul S. Vanderveen, Feb 09 2020

A058934 Numbers k such that 2*5^k + 1 is prime.

Original entry on oeis.org

0, 1, 3, 13, 45, 105, 159, 297, 1443, 2977, 3699, 11709, 12357, 43165, 121995

Offset: 1

Labos Elemer, Jan 12 2001

a(n) is odd for n > 0. Next term > 17200. - Julián Aguirre, Apr 22 2011

a(16) > 2*10^5. - Robert Price, Mar 14 2015

			a(3)=13 and 1 + 2*5^13 = 2441406251 is prime.

Cf. A003306 (2*3^n + 1 is prime), A120375 (2*5^n-1 is prime).

Select[Range[0, 16000], PrimeQ[2*5^#+1]&] (* Julián Aguirre, Apr 22 2011 *)

is(n)=ispseudoprime(2*5^n+1) \\ Charles R Greathouse IV, Jun 12 2017

[n for n in range(1500) if is_prime(2*5^n+1)]
# Julián Aguirre, Apr 22 2011

Edited by Joerg Arndt, Apr 22 2011
Terms 1443,...,12357 from Julián Aguirre, Apr 22 2011
a(14)-a(15) from Robert Price, Mar 14 2015

A216889 Numbers k such that 12*3^k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 5, 13, 14, 38, 200, 248, 884, 1004, 1253, 1634, 3305, 3521, 9601, 19784, 72697

Offset: 1

Vincenzo Librandi, Sep 26 2012

a(19) > 2*10^5. - Robert Price, Mar 16 2014

All terms are verified primes (i.e., not merely probable primes). - Robert Price, Mar 16 2014

Cf. A003306, A005537-A005539, A056805.

[n: n in [0..4000] | IsPrime(12*3^n + 1)];

Select[Range[4000], PrimeQ[12 * 3^# + 1] &]

is(n)=ispseudoprime(12*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A005537(n+1) - 1. - Bruno Berselli, Sep 27 2012

a(16)-a(17) from Vincenzo Librandi, Sep 30 2012

a(18) from Robert Price, Mar 16 2014

cp's OEIS Frontend

A003307 Numbers k such that 2*3^k - 1 is prime.

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A079363 Primes of the form 2*3^k - 1.

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A111974 Primes of the form 2*3^k + 1.

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A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

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A239676 Least k such that k*3^n+1 is prime.

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Magma

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A305237 Numbers m such that m, m+1 and m+2 all have primitive roots.

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A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists.

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A056802 Numbers k such that 2*9^k + 1 is prime.

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Mathematica

PARI

Extensions

A058934 Numbers k such that 2*5^k + 1 is prime.