A005537 -id:A005537 - cp's OEIS frontend

A125734 Primes of the form 4*3^k + 1.

5, 13, 37, 109, 2917, 19131877, 57395629, 16210220612075905069, 3187367866510497232065375864429355521950801431840733951694899540869109890815626195932616388528013, 254244997489062154119688681828370010268347235132197783249391539881181660045297550875174703528321187968562717038040968333

Offset: 1

David Eppstein, Feb 06 2007, Feb 07 2007

nonn

Venkataraman showed that, for every p of this form, 3p is a perfect totient number (cf. A082897).

			37 = 4*3^2 + 1 is a prime of this form. 973 = 4*3^5 + 1 = 7*139 is not a prime, so is not included in this sequence.

T. Venkataraman, Perfect totient number, The Mathematics Student, Vol. 43 (1975), p. 178. MR0447089.

Amiram Eldar, Table of n, a(n) for n = 1..14
Paul Loomis, Michael Plytage and John Polhill, Summing up the Euler phi function, The College Mathematics Journal, Vol. 39, No. 1 (Jan. 2008), pp. 34-42 (see Corollary 3).

Cf. A005537, A082897.

Do[p = 4*3^i + 1; If[PrimeQ@p, Print@p], {i, 0, 300}] (* Robert G. Wilson v, Feb 20 2007 *)

4*3^k + 1 where k belongs to A005537.

2 more terms from Robert G. Wilson v, Feb 20 2007

A245241 Integers n such that 6 * 7^n + 1 is prime.

Original entry on oeis.org

0, 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572

Offset: 1

Robert Price, Nov 14 2014

All terms correspond to verified primes, that is, not merely probable primes.

a(14) > 2*10^5.

			4 is in this sequence because 6 * 7^4 + 1 = 14407, which is prime.

Cf. A003307, A002235, A046865, A079906, A001771, A005541, A056725, A046867, A079907.

Cf. A005537, A005538, A005539, A216889, A216890, A247260.

Select[Range[0,200000], PrimeQ[6 * 7^# + 1] &]

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

A216890 Numbers n such that 14*3^n + 1 is prime.

Original entry on oeis.org

1, 2, 3, 18, 22, 26, 27, 33, 39, 57, 62, 94, 145, 246, 390, 398, 402, 571, 690, 906, 1062, 1254, 1367, 1627, 1954, 2409, 3107, 14754, 15378, 24219, 46138, 98883, 161178

Offset: 1

Vincenzo Librandi, Sep 26 2012

nonn

The next terms are > 6000.
a(34) > 2*10^5. - Robert Price, Mar 16 2014
All terms are verified primes (i.e., not probable primes). - Robert Price, Mar 16 2014

Cf. A003306, A005537-A005539, A056805.

[n: n in [0..4000] | IsPrime(14*3^n+1)];

Select[Range[4000], PrimeQ[14 3^# + 1] &]

is(n)=ispseudoprime(14*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(28)-a(33) from Robert Price, Mar 16 2014

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

A387060 Numbers k such that 16 * 3^k + 1 is prime.

Original entry on oeis.org

0, 3, 4, 5, 12, 24, 36, 77, 195, 296, 297, 533, 545, 644, 884, 932, 1409, 2061, 2453, 2985, 3381, 4980, 5393, 11733, 13631, 14516, 21004, 27663, 32645, 39453, 67055, 90543

Offset: 1

Ken Clements, Aug 15 2025

a(33) > 10^5.
Conjecture: The only intersection with A385115 is at k = 3 where 2^4 * 3^3 = 432 = A027856(8).
Idea: For odd k > 3, covering systems ensure mutual exclusion:
 If k = 1, 9, 13, 19, 25, 31, 37, 39, 43, 49, 55 (mod 60), then 7 or 31 divides (16*3^k+1).
 If k = 5, 7, 11, 17, 23, 27, 29, 35, 41, 47, 53, 57, 59 (mod 60), then 11 or 13 divides (16*3^k-1).
 If k = 15, 21, 33, 45, 51 (mod 60), various primes including {11,31,43,109,277,433,...} ensure at least one of 16*3^k +- 1 is composite.
 If k = 3 (mod 60) and k > 3, the probability of intersection becomes vanishingly small.
 Only k = 3 escapes all divisibility conditions. Verified to k = 10^5.

Ken Clements, Python program to calculate covering system.

Cf. A027856, A005537, A005538, A385115.

Select[Range[0, 4000], PrimeQ[16*3^# + 1] &] (* Amiram Eldar, Aug 16 2025 *)

from gmpy2 import is_prime
print([k for k in range(4_000) if is_prime(16 * 3**k + 1)])

A216888 Numbers k such that 6*3^k + 1 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 15, 16, 29, 53, 56, 59, 64, 131, 179, 319, 695, 781, 821, 896, 1251, 1453, 4216, 5479, 6224, 7841, 12095, 13781, 17719, 43955, 64821, 82779, 105105, 152528, 165895, 191813, 529679, 1074725, 1086111, 1175231, 1277861, 1346541, 3123035, 3648968, 5570080, 6236771, 10852676

Offset: 1

Vincenzo Librandi, Sep 26 2012

			3 is a term because 6*3^3 + 1 = 163 is prime.
7 is not a term because 6*3^7 + 1 = 13123 = 11*1193 is composite.

Cf. A003306, A005537, A005538, A005539; A056800, A056805, A143279.

Associated primes are in A111974.

/* Gives only the terms up to 1453: */ [n: n in [0..1500] | IsPrime(6*3^n + 1)];

Select[Range[5000], PrimeQ[6 3^# + 1] &]

is(n)=ispseudoprime(6*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A003306(n+1)-1. - Bruno Berselli, Sep 27 2012

More terms from Vincenzo Librandi, Oct 01 2012

a(41)-a(47) from the data at A003306 added by Amiram Eldar, Jul 18 2025

A387201 Numbers k such that 32 * 3^k + 1 is prime.

Original entry on oeis.org

1, 4, 8, 9, 32, 36, 48, 74, 112, 186, 204, 364, 393, 572, 781, 1208, 2624, 2778, 4522, 4896, 5272, 32884

Offset: 1

Ken Clements, Aug 21 2025

a(23) > 10^5.

Conjecture: This sequence intersects with A387197 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k = 4(mod 60), and for k > 4 with k = 4(mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.

Ken Clements, Python program to calculate covering system.

Cf. A027856, A003306, A005537, A005538, A387060, A387197.

Select[Range[0, 5000], PrimeQ[32 * 3^# + 1] &] (* Amiram Eldar, Aug 21 2025 *)

from gmpy2 import is_prime
print([ k for k in range(4000) if is_prime(32 * 3**k + 1)])

cp's OEIS Frontend

A125734 Primes of the form 4*3^k + 1.

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A245241 Integers n such that 6 * 7^n + 1 is prime.

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

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A216890 Numbers n such that 14*3^n + 1 is prime.

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A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

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PARI

A387060 Numbers k such that 16 * 3^k + 1 is prime.

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Python

A216888 Numbers k such that 6*3^k + 1 is prime.

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Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A387201 Numbers k such that 32 * 3^k + 1 is prime.

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