A003306 -id:A003306 - cp's OEIS frontend

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

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

A249903 Numbers n such that 2n+1 and sigma(n) are both noncomposite numbers.

Original entry on oeis.org

1, 2, 9, 729

Offset: 1

Jaroslav Krizek, Nov 14 2014

If a(5) exists, it must be a square bigger than 3*10^8.
Intersection of A005097 and A023194.
Conjecture: 2 and 9 are the only numbers n such that 2n - 1, 2n + 1 and sigma(n) are all primes.
From Hiroaki Yamanouchi, Nov 19 2014: (Start)
a(n) (n >= 3) must be of the form 3^(2k) for some positive integer k.
a(5) (if it exists) >= 3^877000 (see A003306 and A028491).
(End)

			Number 729 is in the sequence because 2*729 + 1 = 1459 and sigma(729) = 1093 (both primes).

Cf. A000203, A003306, A005097, A023194, A028491, A249902.

[1] cat [n: n in [1..10000000] | IsPrime(2*n+1) and IsPrime(SumOfDivisors(n))]; // corrected by Vincenzo Librandi, Nov 14 2014

Join[{1}, Select[Range[0, 1000], PrimeQ[DivisorSigma[1, #]]&& PrimeQ[2 # + 1] &]] (* Vincenzo Librandi, Nov 14 2014 *)
Join[{1},Select[Range[1000],AllTrue[{2#+1,DivisorSigma[1,#]},PrimeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 06 2019 *)

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)))

A342037 Numbers k such that A307437(k) is divisible by 3.

Original entry on oeis.org

1, 27, 702, 1107, 1431, 2187, 3375, 3456, 4266, 5157, 5805, 6561, 6831, 7668, 8073, 11313, 11961, 12771, 12825, 13149, 13176, 13257, 14526, 14715, 14796, 15039, 16011, 16227, 16497, 17388, 17496, 17631, 19251, 19332, 19413, 20223, 20277, 20871, 20952, 21654

Offset: 1

Jianing Song, Feb 26 2021

nonn

Indices of terms of A307437 that is divisible by 3.
For e > 0, 3^e is a term if and only if 2*3^e+1 is composite. Hence this sequence is infinite.
All terms > 1 are divisible by 27. Proof: Write a term k = 3^a*r > 1, 3 does not divide r. Suppose A307437(k) = 3^e*s, 3 does not divide s, e >= 1.
i) a = 0. If s = 1, then 2k = 2r divides psi(3^e) = 2*3^(e-1) => k = r = 1, a contradiction. Hence s > 1, then 2k = 2r divides psi(s).
ii) a = 1. If s has a prime factor congruent to 1 modulo 3, then r | psi(s) => 2k = 6r divides psi(s). Otherwise, we must have 3 | psi(3^e) => e >= 2, then 2k = 6r divides psi(7*s), a contradiction.
iii) a = 2. If s has a prime factor congruent to 1 modulo 9, then r | psi(s) => 2k = 18r divides psi(s). Otherwise, we must have 9 | psi(3^e) => e >= 3, then 2k = 18r divides psi(19*s), a contradiction.

			The smallest k such that 2*702 | psi(k) is k = 4293 = 3^4 * 53, hence 702 is a term.
The smallest k such that 2*3375 | psi(k) is k = 20331 = 3^4 * 251, hence 3375 is a term.
The smallest k such that 2*3456 | psi(k) is k = 20817 = 3^4 * 257, hence 3456 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A307437, A341887, A003306.

print1("1, "); forstep(n=27, 10000, 27, if(A307437(n)%3==0, print1(n, ", "))) \\ see A307437 for its program

More terms from Chai Wah Wu, Feb 27 2021

A120491 Numbers k such that 22*3^k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 12, 14, 24, 34, 37, 52, 56, 65, 68, 96, 106, 128, 156, 169, 236, 254, 481, 618, 641, 672, 700, 774, 1274, 1625, 1841, 4650, 6030, 8372, 8760, 9173, 12776, 13873, 25214, 25548, 33834, 72005

Offset: 1

Parthasarathy Nambi, Aug 04 2006

			If k=96 then 22*3^k + 1 is a prime with 48 digits.

Cf. A003306.

[ n: n in [0..3000] | IsPrime(22*3^n + 1) ]; // Vincenzo Librandi, Jan 31 2011

Select[Range[0, 2000], PrimeQ[22*3^# + 1] &] (* Stefan Steinerberger, Aug 06 2006 *)

is(n)=ispseudoprime(22*3^n+1) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Stefan Steinerberger, Aug 06 2006
a(32)-a(41) from Michael S. Branicky, Jul 14 2023
a(42) from Michael S. Branicky, Oct 24 2024

A136041 Largest prime p such that phi^n(p) = 2, where phi^n means n iterations of Euler's totient function.

Original entry on oeis.org

3, 7, 19, 43, 163, 487, 1459, 3079, 8803, 39367, 78787, 196831, 581743, 2125819, 6381667, 19131877, 86093443, 258280327, 516560659, 1214874127

Offset: 1

T. D. Noe, Dec 12 2007

The largest prime in row n+1 of A058812. From Shapiro, we know that a(n) <= 1 + 2*3^(n-1). This bound is attained for n=1,2,3,5,6,7,17,18,.., which is n=A003306(k)+1 for k=1,2,3,...

Harold Shapiro, An arithmetic function arising from the phi function, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.

nn=20; pk=Table[0,{nn}]; Do[p=Prime[n]; k=Length[NestWhileList[EulerPhi,p,#>2&]]-1; If[0

A176351 Numbers n such that 2*3^n + 1 is a primitive prime factor of 10^3^n - 1.

Original entry on oeis.org

4, 180, 320, 5480, 12096, 17720, 82780, 1175232

Offset: 1

T. D. Noe, Apr 15 2010

Consider the problem of finding the smallest number k such that the decimal representation of 1/k has period 3^e for a given e. The number k is usually 3^(e+2). However, if e is one of the n in this sequence, then the prime 2*3^n+1 is a smaller k. The first instance of these exceptions is 1/163, which has a period of 81.
Subsequence of A003306.
10 must be a square residue modulo 2*3^n + 1, implying that n must be a multiple of 4.

Cf. A003306 (primes of the form 2*3^n+1), A003060 (least k such that 1/k has period n).

Select[Range[10000], PrimeQ[1+2*3^# ] && MultiplicativeOrder[10,1+2*3^# ] == 3^# &]

Two more terms from Max Alekseyev, May 03 2010

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

A326655 Numbers k such that 3*4^k+1 is prime.

Original entry on oeis.org

1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, 1104, 1408, 1584, 1956, 17175, 21147, 24075, 27396, 27591, 40095, 354984, 400989, 916248, 1145805, 2541153, 5414673

Offset: 1

Richard N. Smith, Jul 16 2019

One half of the even terms in A002253.

Numbers k such that r*(r+1)^k+1 is prime: A003306 (r=2), this sequence (r=3), A204322 (r=4), A247260 (r=5), A245241 (r=6), A269544 (r=7), A056799 (r=8), A056797 (r=9), A057462 (r=10), A251259 (r=11).

A334993 Numbers k such that 2*3^k + 1 is prime and divides Phi(3^m, 2).

Original entry on oeis.org

1, 5, 9, 17, 57, 65, 897, 4217, 6225, 152529, 3648969, 5570081

Offset: 1

Serge Batalov, May 18 2020

A subset of odd values from A003306.
If p = 2*3^k + 1 is prime then p divides 2^(3^k) + (-1)^k, due to Euler's criterion.
Only odd terms of sequence A003306 can divide the cyclotomic expression Phi(3^m, 2); none of the even terms of sequence A003306 can divide 2^3^k-1 and therefore cannot divide Phi(3^m, 2).

C. Caldwell's Prime Page for Divides Phi category.

Cf. A003306.

dp(n)=Mod(2,2*3^n+1)^3^n==1;
forstep(n=1,6225,2,if(dp(n),print1(n,", ")))

cp's OEIS Frontend

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

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A249903 Numbers n such that 2n+1 and sigma(n) are both noncomposite numbers.

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

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A342037 Numbers k such that A307437(k) is divisible by 3.

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A120491 Numbers k such that 22*3^k + 1 is prime.

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A136041 Largest prime p such that phi^n(p) = 2, where phi^n means n iterations of Euler's totient function.

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A176351 Numbers n such that 2*3^n + 1 is a primitive prime factor of 10^3^n - 1.

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Mathematica

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

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