A002235 -id:A002235 - cp's OEIS frontend

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

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

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)))

A175171 Primes p such that 3*2^p-1 is also prime.

Original entry on oeis.org

2, 3, 7, 11, 43, 103, 827, 7559, 26459, 164987

Offset: 1

Vincenzo Librandi, Mar 09 2010

These are the primes in A002235. [R. J. Mathar, May 02 2010]

			For p=2, 3*2^2-1=11; p=3, 3*2^3-1=23; p=7, 3*2^7-1=383

Cf. A002235.

[p: p in PrimesUpTo(7000) | IsPrime(3*2^p-1) ];

3 more terms from R. J. Mathar, May 02 2010

A175541 A007505 in binary.

Original entry on oeis.org

10, 101, 1011, 10111, 101111, 10111111, 101111111, 1011111111111, 10111111111111111111, 101111111111111111111111111111111111, 1011111111111111111111111111111111111111

Offset: 1

Juri-Stepan Gerasimov, Jun 24 2010

Primes of the form 10, 101, 1011, 10111,..

			a(12)=101111111111111111111111111111111111111111111.

FromDigits/@Select[Table[PadRight[{1,0},n,1],{n,0,50}], PrimeQ[ FromDigits[ #,2]]&] (* Harvey P. Dale, Nov 12 2011 *)

A002235(n)=A091997(n)-A000012(n)=A000120(A007505(n))-A023416(A007505(n)).

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

A267943 Numbers n such that 2^n - 3 and 3*2^n - 1 are both prime.

Original entry on oeis.org

3, 4, 6, 94

Offset: 1

Arkadiusz Wesolowski, Jan 22 2016

The intersection of A002235 and A050414 is not empty (3 does not belong to A267985).

			a(3) = 6 because 2^6 - 3 = 61 and 3*2^6 - 1 = 191 are both prime.

Cf. A002235, A007505, A050414, A050415, A238694, A267985.

[n: n in [2..94] | IsPrime(2^n-3) and IsPrime(3*2^n-1)];

isok(n) = isprime(2^n-3) && isprime(3*2^n-1);

A002235 INTERSECT A050414.

A275247 a(n) = number of decimal digits of A007505(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 6, 11, 12, 14, 18, 20, 24, 29, 32, 44, 63, 66, 93, 99, 119, 139, 142, 250, 384, 987, 1267, 1546, 2276, 3817, 4486, 5457, 5666, 7734, 7966, 12532, 15470, 21610, 24183, 25795, 26543, 29220, 37217, 46941, 49667, 70671, 124880, 176102, 211335, 219060, 298833, 361552, 370947, 696203, 944108, 1274988, 1833429, 3457035, 3531640, 3580969

Offset: 1

Zak Seidov, Jul 21 2016

Cf. A055642, A007505 (primes of form 3*2^n-1), A002235 (numbers n such that 3*2^n-1 is prime).

a(n) = A055642(A007505(n)). - Michel Marcus, Jul 23 2016

cp's OEIS Frontend

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

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

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

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A175171 Primes p such that 3*2^p-1 is also prime.

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A175541 A007505 in binary.

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

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A267943 Numbers n such that 2^n - 3 and 3*2^n - 1 are both prime.

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A275247 a(n) = number of decimal digits of A007505(n).

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