A059609 -id:A059609 - cp's OEIS frontend

A057220 Numbers k such that 2^k - 23 is prime.

2, 4, 6, 8, 12, 14, 18, 36, 68, 152, 212, 324, 1434, 1592, 1668, 3338, 7908, 9662, 27968, 28116, 33974, 41774, 66804, 144518, 162954, 241032, 366218, 676592, 991968

Offset: 1

Robert G. Wilson v, Sep 16 2000

Note that for the values 2 and 4 the primes are negative.
a(22) > 41358. - Jinyuan Wang, Jan 20 2020
All terms are even. - Elmo R. Oliveira, Nov 24 2023

			k = 6: 2^6 - 23 = 41 is prime.
k = 8: 2^8 - 23 = 233 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-23, PRP Top Records.

Cf. A096502.

Cf. 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), this sequence (d=23), A356826 (d=29).

Do[ If[ PrimeQ[ 2^n - 23 ], Print[ n ] ], { n, 1, 15000} ]

is(n)=ispseudoprime(2^n-23) \\ Charles R Greathouse IV, Jun 13 2017

a(19)-a(21) from Jinyuan Wang, Jan 20 2020

a(22)-a(23) found by Henri Lifchitz, a(24)-a(27) found by Lelio R Paula, a(28)-a(29) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 24 2023

A356826 Numbers k such that 2^k - 29 is prime.

Original entry on oeis.org

5, 8, 104, 212, 79316, 102272, 225536, 340688

Offset: 1

Craig J. Beisel, Aug 29 2022

A particularly low-density pseudo-Mersenne sequence. I have verified that there are no additional terms for k < 5*10^4. For k = a(5), a(6), a(7), and a(8), 2^k - 29 is a probable prime (see link).
The terms a(5)-a(8) were discovered by Henri Lifchitz (see link). - Elmo R. Oliveira, Nov 29 2023
Empirically: except for 5, all terms are even. - Elmo R. Oliveira, Nov 29 2023

			5 is a term because 2^5 - 29 = 3 is prime.
8 is a term because 2^8 - 29 = 227 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-29, PRP Top Records.

Cf. A096502.

Cf. 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), this sequence (d=29).

for(n=2, 1000, if(isprime(2^n-29), print1(n, ", ")))

A088834 Numbers k such that sigma(k) == 6 (mod k).

Original entry on oeis.org

1, 5, 6, 25, 180, 8925, 32445, 442365

Offset: 1

Labos Elemer, Oct 29 2003

For each integer j in A059609, 2^(j-1)*(2^j - 7) is in the sequence. E.g., for j = A059609(1) = 39 we get 151115727449904501489664. - M. F. Hasler and Farideh Firoozbakht, Dec 03 2013
No more terms to 10^10. - Charles R Greathouse IV, Dec 05 2013
a(9) > 10^13. - Giovanni Resta, Apr 02 2014
a(9) > 1.5*10^14. - Jud McCranie, Jun 02 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			Sigma(25) = 31 = 1*25 + 6, so 31 mod 25 = 6.

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. Article ID 10.3.1.

Cf. A054024, A045768, A045769, A045770, A077374, A076496, A088012.
Cf. A087167 (a subsequence).
Cf. A059609.

Select[Range[1000000], Mod[DivisorSigma[1, #] - 6, #] == 0 &] (* T. D. Noe, Dec 03 2013 *)

isok(n) = Mod(sigma(n), n) == 6; \\ Michel Marcus, Jan 03 2023

Terms corrected by Charles R Greathouse IV and Farideh Firoozbakht, Dec 03 2013

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

A217380 Numbers k such that 8^k - 7 is prime.

Original entry on oeis.org

13, 661, 773, 833, 4273, 40613, 302913

Offset: 1

Vincenzo Librandi, Oct 02 2012

All terms are equal to 1/3 of the multiples of 3 in A059609.

Naturally these numbers are odd since (9-1)^(2n)-7 is divisible by 3. - Bruno Berselli, Oct 04 2012

Cf. A059609, A217353, A217356.

Mathematica
```
Select[Range[4000], PrimeQ[8^# - 7] &]
```

is(n)=ispseudoprime(8^n-7) \\ Charles R Greathouse IV, Jun 13 2017

a(7) using A059609 from Michael S. Branicky, Sep 15 2024

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

A269835 Primes p of the form 2^k + 4*(-1)^k - 3.

Original entry on oeis.org

2, 5, 17, 257, 65537, 549755813881

Offset: 1

Jaroslav Krizek, Mar 06 2016

nonn

a(7) has 216 digits (see b-file).
Fermat primes > 3 from A019434 are terms.
Corresponding values of k: 0, 2, 4, 8, 16, 39, 715, ....
Note that for k = 1, 2^k + 4*(-1)^k - 3 = -5.
For further k values, see A059609. (2^k-7 is divisible by 3 for even k.) - Jeppe Stig Nielsen, Nov 18 2019

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10 (terms 1..7 from Jaroslav Krizek).

Cf. A019434, A059609.

[2] cat [2^k + 4*(-1)^k - 3: k in [2..300] | IsPrime(2^k + 4*(-1)^k - 3)];

Select[Table[2^k+4(-1)^k-3,{k,0,50}],Positive[#]&&PrimeQ[#]&] (* Harvey P. Dale, Sep 14 2019 *)

for (k=0,40,my(j=2^k+4*(-1)^k-3);if(isprime(j),print1(j,", "))) \\ Hugo Pfoertner, Nov 21 2019

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

cp's OEIS Frontend

A057220 Numbers k such that 2^k - 23 is prime.

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A356826 Numbers k such that 2^k - 29 is prime.

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A088834 Numbers k such that sigma(k) == 6 (mod k).

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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.

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A217380 Numbers k such that 8^k - 7 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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Magma

Mathematica

PARI

A269835 Primes p of the form 2^k + 4*(-1)^k - 3.

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Author

Keywords

Comments

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Programs

Magma

Mathematica

PARI

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

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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.