A057220 -id:A057220 - cp's OEIS frontend

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

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, ", ")))

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

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

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