A096819 -id:A096819 - 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, ", ")))

A223610 Numbers k whose abundance is 18: sigma(k) - 2*k = 18.

Original entry on oeis.org

208, 6976, 8415, 31815, 351351, 2077696, 20487159, 159030135, 536559616, 2586415095, 137433972736, 2199003332608, 2305842988812599296

Offset: 1

Donovan Johnson, Mar 23 2013, at suggestion of N. J. A. Sloane and Robert G. Wilson v

Any term x of this sequence can be combined with any term y of A223608 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016

Every number of the form 2^(j-1)*(2^j - 19), where 2^j - 19 is prime, is a term (cf. A096819). - Jon E. Schoenfield, Jun 02 2019

			For k = 159030135, sigma(k) - 2*k = 18.

Cf. A000203, A033880, A096819, A223608 (deficiency 18).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 18]; // Vincenzo Librandi, Sep 14 2016

Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == 18 &] (* Vincenzo Librandi, Sep 14 2016 *)

for(n=1, 10^8, if(sigma(n)-2*n==18, print1(n ", ")))

a(12) from Giovanni Resta, Mar 29 2013

a(13) from Jon E. Schoenfield confirmed and added by Max Alekseyev, Jun 03 2025

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

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

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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A223610 Numbers k whose abundance is 18: sigma(k) - 2*k = 18.

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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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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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Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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.