A050414 -id:A050414 - cp's OEIS frontend

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

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)))

A321353 Numbers k such that 3*2^k - 25 is prime.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 12, 13, 16, 17, 18, 21, 23, 29, 31, 33, 35, 36, 41, 58, 63, 66, 69, 82, 96, 99, 148, 157, 175, 196, 241, 267, 349, 394, 404, 414, 435, 456, 485, 498, 537, 548, 584, 715, 727, 765, 929, 1007, 1076, 1399, 1619, 1652, 1715, 2758, 3039, 3131, 3773, 3822, 5001

Offset: 1

Gilbert Mozzo, Nov 07 2018

nonn

Appears (at least initially) to contain more primes that the analogous sequences for 2^k-1 or 2^k-3. Compare the comment of Paul Bourdelais in A050414.

			7 is a term, because 3*2^7 - 25 = 359 is prime.

Cf. A050414.

Select[Range[100], PrimeQ[3 2^# - 25] &]

for(n=0, 2000, if(ispseudoprime(p=3*2^n-25), print1(n, ", ")));

A329103 Numbers k such that both k^2 - 3 and 2^k - 3 are primes.

Original entry on oeis.org

4, 10, 14, 20, 266, 452, 694

Offset: 1

Alex Ratushnyak, Nov 04 2019

a(8) > 2086750, using A050414. - Michael S. Branicky, Feb 16 2024

Intersection of A050414 and A028873.

Cf. A000040, A329102.

isok(k) = isprime(k^2 - 3) && isprime(2^k - 3); \\ Michel Marcus, Jul 02 2021

A331487 Primes p such that exactly one of 2^(p+1) - 3 and 2^(p+1) + 3 is a prime.

Original entry on oeis.org

13, 17, 19, 23, 29, 83, 149, 173, 227, 389, 1109, 4001, 35753, 36551, 363119, 702193

Offset: 1

Juri-Stepan Gerasimov, Jan 18 2020

Primes p such that exactly one of k*2^p - 2*k + 1 and k*2^p + 2*k - 1 is a prime:
k = 1: odd terms in A000043;
k = 2: this sequence;
k = 3: 5, 13, 19, 29, 31, 109, 139, 271, 379, 1553, ...
k = 4: 2, 37, ...
k = 5: 3, 5, 7, 17, 19, 23, 41, 61, 67, 151, 157, 313, 4111, 6337, ...
k = 6: 2, 5, 7, 11, 19, 29, 149, 191, 373, 449, 983, 1667, 1973, ...
k = 7: 2, 3, 5, 7, 11, 13, 29, 43, 61, 97, 109, 127, 131, 239, 461, 1153, ...
k = 8: 3, 11, 19, 23, 29, 37, 43, 97, 193, 307, 617, 1847, ...
k = 9: 3, 5, 23, 41, 61, 71, 97, 131, 157, 863, 3119, ...
k = 10: 2, 3, 13, ...
...

			13 is in this sequence because 2^(13+1) - 3 = 16381 (prime) and 2^(13+1) + 3 = 16387 (composite number).

Cf. A000043, A050414, A057732, A174269.

[p: p in PrimesUpTo(1000) | not (#[k: k in [2] | IsPrime(k*2^p-2*k+1)]) eq (#[k: k in [2] | IsPrime(k*2^p+2*k-1)])];

Select[Range[400], PrimeQ[#] && Xor @@ PrimeQ[2^(# + 1) + {-3, 3}] &] (* Amiram Eldar, Jan 19 2020 *)

isok(p) = isprime(2*2^p-3) + isprime(2*2^p+3) == 1;
forprime(p=2, 500, if(isok(p), print1(p, ", "))); \\ Jinyuan Wang, Jan 19 2020

a(12)-a(16) added using A050414 and A057732 by Jinyuan Wang, May 15 2020

A379020 Numbers k such that 2^k - 25 is prime.

Original entry on oeis.org

5, 7, 9, 13, 33, 37, 57, 63, 93, 127, 129, 165, 189, 369, 717, 3079, 3087, 3925, 6709, 7633, 18001, 21961, 55557, 60415, 63589, 69463, 75949, 98265, 212295, 416773, 647545, 824325, 1538959, 2020893, 2421175

Offset: 1

Boyan Hu, Dec 13 2024

Except for a(1), all terms are congruent to 1 or 3 mod 6.

a(36) > 3400000. - Boyan Hu, Jun 16 2025

			7 is in the sequence because 2^7-25=103 is prime.
8 is not in the sequence because 2^8-25=231=3*7*11 is not prime.

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-25, PRP Top Records.
Boyan Hu, 2^n-25 PRP searching progress

Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Except for a(1), subsequence of A047241.

Do[ If[ PrimeQ[ 2^n - 25 ], Print[ n ] ], { n, 1, 15000} ]

PARI
```
is(n)=ispseudoprime(2^n-25)
```

a(1)=5 inserted by Max Alekseyev, May 28 2025

A174272 Exactly one of 2^n-3 and 2^n+3 is prime.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 14, 15, 16, 18, 20, 22, 24, 28, 29, 30, 55, 67, 84, 94, 116, 122, 150, 174, 213, 221, 228, 233, 266, 336, 390, 452, 545, 689, 694, 784, 850, 1110, 1704, 1736, 2008, 2139, 2191, 2321, 2367, 2370, 3237, 3954, 4002, 4060, 4062, 4552, 5547, 5630

Offset: 1

Juri-Stepan Gerasimov, Mar 14 2010

nonn

Numbers n which are in A050414 or in A057732 but not in both. [From R. J. Mathar, Mar 29 2010]

			a(1)=1 because 2^1-3=-1 is nonprime and 2^1+3=5 is prime.

Cf. A050414, A057732.

Corrected by Charles R Greathouse IV, Mar 20 2010

A192542 Numbers n such that the interval [2^n-n, 2^n] is prime-free.

Original entry on oeis.org

15, 25, 27, 28, 30, 34, 38, 40, 43, 45, 47, 48, 49, 51, 53, 55, 60, 71, 72, 75, 88, 97, 99, 106, 113, 117, 126, 128, 132, 139, 145, 146, 148, 151, 154, 168, 169, 175, 176, 177, 185, 186, 192, 208, 216, 223, 227, 232, 240, 253, 259

Offset: 1

Juri-Stepan Gerasimov, Jul 03 2011

nonn

Places n where A192064(n)=0.

The numbers not in the sequence are characterized in A000043, A050414, A059608, A059610, etc.

Cf. A192064.

Select[Range[260],NextPrime[2^#-#]>2^#&] (* Harvey P. Dale, Apr 08 2013 *)

is(n)=nextprime(2^n-n)>2^n \\ Charles R Greathouse IV, Jun 13 2017

Corrected (a(16)=55 inserted) by Harvey P. Dale, Apr 08 2013

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.

A321356 Primes of the form 3*2^k - 25.

Original entry on oeis.org

23, 71, 167, 359, 743, 1511, 12263, 24551, 196583, 393191, 786407, 6291431, 25165799, 1610612711, 6442450919, 25769803751, 103079215079, 206158430183, 6597069766631, 864691128455135207, 27670116110564327399, 221360928884514619367, 1770887431076116955111

Offset: 1

Gilbert Mozzo, Nov 07 2018

nonn

Cf. A050414, A321353 (k values).

Select[3*2^Range[4,100]-25 , PrimeQ] (* Amiram Eldar, Nov 15 2018 *)

for(n=0,20, if(ispseudoprime(p=3*2^n-25), print1(p, ", ")))

More terms from Amiram Eldar, Nov 15 2018

cp's OEIS Frontend

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A321353 Numbers k such that 3*2^k - 25 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A329103 Numbers k such that both k^2 - 3 and 2^k - 3 are primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A331487 Primes p such that exactly one of 2^(p+1) - 3 and 2^(p+1) + 3 is a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A379020 Numbers k such that 2^k - 25 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A174272 Exactly one of 2^n-3 and 2^n+3 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A192542 Numbers n such that the interval [2^n-n, 2^n] is prime-free.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI