A059608 -id:A059608 - cp's OEIS frontend

A217356 Numbers n such that 8^n - 5 is prime.

Original entry on oeis.org

1, 2, 4, 6, 12, 22, 50, 270, 606, 800, 1704, 5462, 6194, 9206, 10306, 105380

Offset: 1

Vincenzo Librandi, Oct 02 2012

nonn

All terms are equal to 1/3 of the multiples of 3 in A059608.

Cf. A059608, A217355.

Mathematica
```
Select[Range[4000], PrimeQ[8^# - 5] &]
```

is(n)=ispseudoprime(8^n-5) \\ Charles R Greathouse IV, Jun 06 2017

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

A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

Original entry on oeis.org

11, 251, 1019, 4091, 65531, 4294967291

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

By comparing A059608 and A001770, the next term, if it exists, is larger than 2^350028. - Giovanni Resta, Mar 06 2014
Lesser prime of Mersenne prime pair of order k and of the form {2^m - (2k - 1), (2k - 1)*2^m - 1}:
for order k = 1: 3, 7, 31, 127, 8191, 131071, ... (Mersenne primes A000668)
for order k = 2: 5, 13, 61, ...
for order k = 3: 11, 251, 1019, 4091, 655531, 4294967291, ... (this sequence)
for order k = 4:
for order k = 5: 2097143, ...
for order k = 6: 3, ...
for order k = 7:
for order k = 8: 17, 1009, 16369, ...
for order k = 9: 47, 65519, 1048559, 68719476719, ...
for order k = 10: 13, 2097133, ...
for order k = 11: 107, 8171, ...
for order k = 12: 41, 233, 4073, ...
for order k = 13: 487, ...
for order k = 14: 5, 229, 997, ...
for order k = 15: 97, ...

			11 is in this sequence because Mersenne prime pair {2^4-(2*3-1) = 11, (2*3-1)*2^4-1 = 79} where 3 is order and 11 is lesser prime (for m = 4).

Cf. A237422, A238694, A238749, A059608, A001770, A156560, A050522.

2^Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5*2^# - 1] &] - 5 (* Giovanni Resta, Mar 06 2014 *)

Numbers 2^m - 5 for m belonging to the intersection of A001770 and A059608. - Max Alekseyev, Feb 20 2024

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

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

A344516 Numbers k such that (2^k-5, 2^k-3) are twin primes.

Original entry on oeis.org

3, 4, 6, 10, 12, 20, 150

Offset: 1

Amiram Eldar, May 21 2021

No more terms <= A050414(66) = 2086750.

			3 is a term since (2^3-5, 2^3-3) = (3, 5) are twin primes.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 150.

Intersection of A050414 and A059608.

Cf. A001097.

Select[Range[150], And @@ PrimeQ[2^# - {3,5}] &]

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A217356 Numbers n such that 8^n - 5 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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A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

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

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

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A344516 Numbers k such that (2^k-5, 2^k-3) are twin primes.

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