A002236 -id:A002236 - cp's OEIS frontend

A050524 Primes of the form 9*2^n-1.

17, 71, 1151, 73727, 294911, 18874367, 79164837199871, 83010348331692982271, 5704427701027032306735164424191, 5841333965851681082096808370372607, 6576757367989063131916581747223273588451696443391

Offset: 1

N. J. A. Sloane, Dec 29 1999

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

See A002236 for more terms.

Select[9 2^Range[0,160]-1,PrimeQ]  (* Harvey P. Dale, Apr 13 2011 *)

A230527 Numbers n such that 3^6*2^n - 1 is prime.

Original entry on oeis.org

5, 17, 23, 113, 125, 173, 199, 319, 377, 397, 785, 937, 2167, 3785, 3977, 5957, 7727, 8249, 14677, 19577, 20485, 36319, 57509, 60703, 66677, 76877, 77017, 83407, 229405, 1003373

Offset: 1

Lei Zhou, Oct 22 2013

Riesel Primes with k = 3^6 = 729.

Checked up to n = 1003600.

			729*2^5-1=23327 is a prime number.

K. Bonath, Riesel Prime Database
C. K. Caldwell, 729*2^229405-1
C. K. Caldwell, 729*2^1003373-1

Cf. A002235, A002236, A050539, A050566, A050880.

b=3^6; i=0; Table[While[i++; cp=b*2^i-1; !PrimeQ[cp]]; i, {j, 1, 15}]

is(n)=ispseudoprime(3^6*2^n-1) \\ Charles R Greathouse IV, May 22 2017

Lei Zhou, Nov 08 2013, added a Mathematica program for small elements.

A230537 Numbers n such that 3^7*2^n - 1 is prime.

Original entry on oeis.org

1, 2, 6, 13, 22, 29, 30, 33, 36, 50, 61, 118, 180, 226, 405, 433, 522, 789, 929, 960, 1026, 1030, 1118, 1266, 1521, 1718, 2536, 3029, 3366, 4253, 9157, 10165, 23641, 29877, 30648, 47265, 56097, 90501, 101981, 103021, 108370, 117909, 157237, 169156, 174168

Offset: 1

Lei Zhou, Oct 22 2013

Riesel Primes with k = 3^7 = 2187.

Checked up to n = 1000000.

			2187*2^1-1=4373 is a prime number.

Lei Zhou, Table of n, a(n) for n = 1..52
K. Bonath, Riesel Prime Database
C. K. Caldwell, a(51) = 2187*2^449776-1
C. K. Caldwell, a(52) = 2187*2^916686-1
C. K. Caldwell, a(53) = 2187*2^992632-1

Cf. A002235, A002236, A050539, A050566, A050880, A230527.

b=3^7;i=0; Table[While[i++; cp=b*2^i-1; !PrimeQ[cp]]; i, {j, 1, 30}]

is(n)=ispseudoprime(3^7*2^n-1) \\ Charles R Greathouse IV, May 22 2017

Lei Zhou, Nov 08 2013, added a Mathematica program for small elements.

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.

Original entry on oeis.org

0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 05 2014

nonn

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:
For k = 0: 2, 3, 5, 7, 13, 17, ...   Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ...          Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3:                           Intersection of A059609 and A001771
for k = 4: 21, ...                   Intersection of A059610 and A002236
for k = 5:                           Intersection of A096817 and A001772
for k = 6:                           Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ...            Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ...        Intersection of A059611 and A001774
for k = 9: 5, 21, ...                Intersection of A096819 and A001775
for k = 10: 7, 13, ...               Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...

			a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).

a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)

a(0), a(19), a(20) corrected by Giovanni Resta, Mar 13 2014

A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

Original entry on oeis.org

34, 568, 147328, 603971584, 9663643648, 39582416502784, 696341272098017608537735168, 765635325572111542783369494684623699968, 3615610599582728119969414707766982030374842621310535527825408, 3791242500068058721125048996612134914443116117566314438843154038784

Offset: 1

Jinyuan Wang, Feb 12 2020

nonn

This is the case h = 8 of the h-perfect numbers as defined in the Harborth link.

			34 is a term since sigma(34) + sigma(8*34) = 612, that is 18*34.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A002236, A050524, A136539, A281993, A330413.

isok(m) = sigma(m) + sigma(8*m) == 18*m;

a(n) = 2^A002236(n) * A050524(n).

A212311 Numbers n such that 3^8*2^n - 1 is prime.

Original entry on oeis.org

1, 15, 33, 43, 61, 121, 295, 315, 367, 681, 771, 789, 1485, 4915, 5305, 33649, 81343, 85005, 116307, 154869, 230731, 279591, 287847, 329515, 545353, 1053481

Offset: 1

Lei Zhou, Oct 24 2013

Riesel Primes with k = 3^8 = 6561.

Checked up to n = 1053627.

			6561*2^1-1=13121 is a prime number.

K. Bonath,Riesel Prime Database
C. K. Caldwell, 6561*2^1053481-1

Cf. A002235, A002236, A050539, A050566, A050880, A230527, A230537.

b=2^8;i=0; Table[While[i++; cp=b*2^i-1; !PrimeQ[cp]]; i, {j, 1, 13}]

is(n)=ispseudoprime(3^8*2^n-1) \\ Charles R Greathouse IV, Jun 13 2017

Lei Zhou, Nov 08 2013, added a Mathematica program for small elements.

A238749 Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.

Original entry on oeis.org

4, 8, 10, 12, 18, 32

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

a(7) > 350028.
Intersection of A059608 and A001770.
Exponents of Mersenne prime pairs {2^n - (2k + 1), (2k + 1)*2^n - 1}:
for k = 0: 2, 3, 5, 7, 13, 17, ...   Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ...          Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3:                           Intersection of A059609 and A001771
for k = 4: 21, ...                   Intersection of A059610 and A002236
for k = 5:                           Intersection of A096817 and A001772
for k = 6:                           Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ...            Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ...        Intersection of A059611 and A001774
for k = 9: 5, 21, ...                Intersection of A096819 and A001775
for k = 10: 7, 13, ...               Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...
for k = 14:

			a(1) = 4 because 2^4 - 5 = 11 and 5*2^4 - 1 = 79 are both primes.

Cf. A000043, A001770, A059608, A237422, A238694, A238251, A238751, A238797.

[n: n in [0..100] | IsPrime(2^n-5) and IsPrime(5*2^n-1)]; // Vincenzo Librandi, May 17 2015

fQ[n_] := PrimeQ[2^n - 5] && PrimeQ[5*2^n - 1]; k = 1; While[ k < 15001, If[fQ@ k, Print@ k]; k++] (* Robert G. Wilson v, Mar 05 2014 *)
Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5 2^# - 1] &] (* Vincenzo Librandi, May 17 2015 *)

isok(n) = isprime(2^n - 5) && isprime(5*2^n - 1); \\ Michel Marcus, Mar 04 2014

A231374 Numbers n such that 3^9*2^n - 1 is prime.

Original entry on oeis.org

4, 7, 19, 22, 32, 46, 50, 62, 83, 103, 124, 142, 190, 230, 256, 260, 422, 596, 1084, 2270, 2770, 5366, 5434, 5762, 6826, 9239, 15211, 22556, 58790, 81319, 172510, 225350, 236326, 258592, 445364, 975020

Offset: 1

Lei Zhou, Nov 08 2013

Riesel Primes with k = 3^9 = 19683.

Checked up to n = 1000000.

			19683*2^4-1=314927 is a prime number.

Cf. A002235, A002236, A050539, A050566, A050880, A230527, A230537, A212311.

i=0;Table[While[i++;cp=19683*2^i-1;!PrimeQ[cp]];i,{j,1,20}]

is(n)=ispseudoprime(3^9*2^n-1) \\ Charles R Greathouse IV, Jun 13 2017

A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.

Original entry on oeis.org

1, 3, 7, 43, 63, 211

Offset: 1

Ken Clements, Aug 05 2025

The exponent, k, of 2 must be odd because the exponent, 2, of 3 (where 9 = 3^2) is even and the sum of the exponents of prime factors 2 and 3 must be odd to form a product that is a twin prime average. Of all subsequences of A027856, this is the longest known where the power of 3 is fixed.

Amiram Eldar noted that using A002236 and A002256, we obtain a(7) > 5.6*10^6, if it exists.

			a(1) = 1 because 2*9 = 18 with 17 and 19 prime.
a(2) = 3 because 8*9 = 72 with 71 and 73 prime.
a(3) = 7 because 128*9 = 1152 with 1151 and 1153 prime.
a(4) = 43 because 8796093022208*9 = 79164837199872 with 79164837199871 and 79164837199873 prime.

Intersection of A002236 and A002256.

Cf. A014574, A384530, A027856, A385433, A181490.

q:= k-> (m-> andmap(isprime, [m-1, m+1]))(9*2^k):
select(q, [2*i-1$i=1..111])[];  # Alois P. Heinz, Aug 08 2025

from gmpy2 import is_prime
def is_TPpi2(e2, e3):
    N = 2**e2 * 3**e3
    return is_prime(N-1) and is_prime(N+1)
print([k for k in range(1, 100001, 2) if is_TPpi2(k, 2)])

cp's OEIS Frontend

A050524 Primes of the form 9*2^n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A230527 Numbers n such that 3^6*2^n - 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A230537 Numbers n such that 3^7*2^n - 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A238797 Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A330412 Integers m such that sigma(m) + sigma(8*m) = 18*m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A212311 Numbers n such that 3^8*2^n - 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A238749 Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A231374 Numbers n such that 3^9*2^n - 1 is prime.

Views

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.

A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.