A096818 -id:A096818 - 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

A141545 Numbers k whose abundance is 12: sigma(k) - 2*k = 12.

Original entry on oeis.org

24, 30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

nonn

Numbers k such that sigma(k) = 2k + 12. - Wesley Ivan Hurt, Jul 11 2013
Any term x = a(m) can be combined with any term y = A141549(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2.  Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one.  So far, these two sequences have not produced an amicable pair.  However, if one is ever found, then it will exhibit y-x = 12. - Timothy L. Tiffin, Sep 13 2016
From Tomohiro Yamada, Jan 01 2023: (Start)
6p belongs to this sequence if p > 3 is prime since sigma(6p) = 12(p + 1) = 12p + 12.  Moreover, 2^m * (2^(m+1) - 13) is also a term of this sequence if 2^(m+1) - 13 is prime (m+1 is a term of A096818) since sigma(2^m * (2^(m+1) - 13)) = (2^(m+1) + 1) * (2^(m+1) - 13) = 2^(m+1) * (2^(m+1) - 13) + 12.  So 24, 304, 127744, 33501184, and 8589082624 also belong to this sequence.
Problem: is 54 the only term of this sequence which is of neither type given above? (End)

			30 is in the sequence since sigma(30) = sigma(2*3*5) = sigma(2)*sigma(3)*sigma(5) = 3*4*6 = 72 = 2(30)+12.  Since this is the second such number whose abundance is 12, a(2) = 30. - _Wesley Ivan Hurt_, Jul 11 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A000203, A005101, A141549 (deficiency 12).

Cf. A076496 (sigma(k) - a*k = 12).

[n: n in [1..1400] | (SumOfDivisors(n)-2*n) eq 12]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n-12,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^4], DivisorSigma[1, #] - 2 # == 12 &] (* Vincenzo Librandi, Sep 14 2016 *)

is(n)=sigma(n)==2*n+12 \\ Charles R Greathouse IV, Feb 21 2017

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, ", ")))

A084306 Numbers x such that sigma(x) mod x = 12 and x is not divisible by 6. Singular solutions mentioned in A076496.

Original entry on oeis.org

121, 304, 127744, 33501184, 8589082624

Offset: 1

Labos Elemer, Jun 11 2003

nonn

If n = P*q, where P is a multiple perfect number and q is prime so that gcd(P,q) = 1, then sigma(n) = kn(q+1). Consequently sigma(n) = knq + kn sigma(n) mod n = kn. Such values of n are regular solutions to this and analogous cases. Here, not these but the additional eccentric solutions are collected. Cf. A076496.
a(6) > 10^11. - Donovan Johnson, Sep 20 2012
If p = 2^k - 13 > 3 is a prime number, then 2^(k-1)*p is a term. This happens for k = 5, 9, 13, 17, 57, 105, 137, 3217, ... (A096818). - Giovanni Resta, Apr 01 2014

			n = 33501184 = 4096*8179; sigma(n) = 2n + 12 = 67002380.

Cf. A076496, A076495, A000396, A045768-A045770, A007691, A096818.

Do[s=Mod[DivisorSigma[1, n], n]; If[IntegerQ[n/100000], Print[{n}]]; If[Equal[s, 12]&&!Equal[Mod[n, 6], 0], Print[n]], {n, 1, 100000000}]

a(5) from Donovan Johnson, Sep 20 2012

A165778 Numbers k such that |2^k - 57| is prime.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 16, 19, 22, 28, 43, 46, 56, 58, 62, 67, 74, 82, 140, 160, 316, 346, 376, 454, 458, 487, 580, 607, 1018, 1579, 1739, 1870, 2006, 3014, 3056, 6962, 7075, 7852, 8207, 9190, 11854, 14816, 23308, 29222, 33808, 40618, 47408, 50843, 58312, 98554

Offset: 1

M. F. Hasler, Oct 11 2009

nonn

If p = 2^k-57 is prime, then 2^(k-1)*p is in A101260, i.e., a solution to sigma(x)-2x = 56 = 2^3*(2^3-1) = 2*A000396(2).

			a(3) = 6 since 2^6-57 = 7 is prime.
For exponents a(1) = 2 and a(2) = 4, we get 2^a(n)-57 = -53 and -41 which are negative, but which are prime in absolute value.

Cf. A096818, A165779, A165780.

[n: n in [1..1100] | IsPrime(2^n-57)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 57]}, {n, 0, 100}], PrimeQ[#[[2]]] &][[All,1]] (* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-57)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

from sympy import isprime, nextprime
def afind(limit):
    k, pow2 = 1, 2
    for k in range(1, limit+1):
        if isprime(abs(pow2-57)):
            print(k, end=", ")
        k += 1
        pow2 *= 2
afind(2100) # Michael S. Branicky, Dec 27 2021

a(36)-a(42) from Altug Alkan, Apr 08 2016
a(43)-a(44) from Michael S. Branicky, Dec 27 2021
a(45)-a(49) from Michael S. Branicky, May 14 2023
a(50) from Michael S. Branicky, Sep 25 2024

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

A165779 Numbers k such that |2^k-993| is prime.

Original entry on oeis.org

1, 4, 6, 10, 14, 17, 26, 29, 54, 62, 77, 121, 344, 476, 1012, 1717, 1954, 2929, 2993, 3014, 3304, 4704, 8882, 24042, 43572, 45722, 54913, 57893, 72566, 74473, 82092, 117302

Offset: 1

M. F. Hasler, Oct 11 2009

If p = 2^k-993 is prime, then 2^(k-1)*p is a solution to sigma(x)-2x = 992 = 2^5*(2^5-1) = 2*A000396(3).

			a(4) = 10 since 2^10-993 = 31 is prime.
For exponents a(1) = 1, a(2) = 4 and a(3) = 6, we get 2^a(n)-993 = -991, -977 and -929 which are negative, but which are prime in absolute value.

Cf. A000396, A096818, A165778, A165780.

[n: n in [1..1100] |IsPrime(2^n-993)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 993]}, {n,0,100}], PrimeQ[#[[2]]] &][[All, 1]] (* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-993)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

from sympy import isprime, nextprime
def afind(limit):
    k, pow2 = 1, 2
    for k in range(1, limit+1):
        if isprime(abs(pow2-993)):
            print(k, end=", ")
        k += 1
        pow2 *= 2
afind(2000) # Michael S. Branicky, Dec 26 2021

a(23) from Altug Alkan, Apr 08 2016
a(24) from Michael S. Branicky, Dec 26 2021
a(25)-a(26) from Michael S. Branicky, Apr 06 2023
a(27)-a(32) from Michael S. Branicky, Sep 25 2024

A165780 Numbers n such that |2^n-16257| is prime.

Original entry on oeis.org

2, 3, 6, 8, 10, 12, 14, 16, 20, 22, 26, 30, 34, 36, 38, 43, 44, 50, 58, 64, 68, 80, 116, 142, 146, 254, 296, 298, 306, 396, 456, 730, 876, 1004, 1006, 1051, 1094, 1776, 1896, 1908, 2502, 2876, 3824, 3882, 4796, 4818, 5006, 5704, 6722, 8467, 9676

Offset: 1

M. F. Hasler, Oct 11 2009

nonn

If p=2^n-16257 is prime, then 2^(n-1)*p is a solution to sigma(x)-2x = 16256 = 2^7*(2^7-1) = 2*A000396(4).

			a(7)=14 since 2^14-16257 = 127 is prime.
For exponents a(1)=2 through a(6)=12, we get negative values for 2^a(k)-16257, which are prime in absolute value.

Cf. A096818, A165778, A165779.

[n: n in [1..1100] |IsPrime(2^n-16257)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 16257]},{n,0,100}], PrimeQ[#[[2]]] &][[All, 1]](* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-16257)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

More terms from Altug Alkan, Apr 08 2016

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A141545 Numbers k whose abundance is 12: sigma(k) - 2*k = 12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A084306 Numbers x such that sigma(x) mod x = 12 and x is not divisible by 6. Singular solutions mentioned in A076496.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A165778 Numbers k such that |2^k - 57| is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A165779 Numbers k such that |2^k-993| is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

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.