A059608 -id:A059608 - cp's OEIS frontend

A096817 Numbers k such that 2^k - 11 is prime.

4, 6, 10, 18, 42, 78, 94, 114, 190, 322, 546, 3894, 10318, 11650, 12474, 20994, 61810, 103882, 296010, 636930, 653638, 926766

Offset: 1

Labos Elemer, Jul 13 2004

All terms are even since for odd k, 2^k - 11 is divisible by 3.

			k = 6: 64 - 11 = 53 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-11, 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), this sequence (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[4,20000],PrimeQ[2^#-11]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)

a(13)-a(16) from Max Alekseyev, a(17)-a(18) from Henri Lifchitz, added by Max Alekseyev, Feb 09 2012
a(19) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(20)-a(22) from Stefano Morozzi, added by Elmo R. Oliveira, Nov 16 2023

A096819 Numbers k such that 2^k - 19 is prime.

Original entry on oeis.org

5, 7, 11, 15, 19, 21, 31, 39, 67, 69, 85, 157, 171, 191, 255, 291, 379, 3669, 4551, 9531, 13119, 14211, 20647, 233965, 337267, 534429, 535415, 816039, 991715

Offset: 1

Labos Elemer, Jul 13 2004

All terms are odd since for even k, 2^k - 19 is divisible by 3.

a(26) > 5*10^5. - Tyler NeSmith, Apr 16 2022

			2^7 - 19 = 128 - 19 = 109, a prime, so 7 is a term of the sequence.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-19, 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), this sequence (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[5,20000],PrimeQ[2^#-19]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011*)

a(22)-a(23) from Max Alekseyev, Feb 10 2012
a(24)-a(25) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(26)-a(29) found by Stefano Morozzi, added by Alois P. Heinz, Aug 29 2022

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

Original entry on oeis.org

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, ", ")))

A045769 Numbers k such that sigma(k) == 4 (mod k).

Original entry on oeis.org

1, 3, 9, 12, 70, 88, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088, 9223372026117357568

Offset: 1

Dan Hoey

Every number of the form 2^(j-1)*(2^j - 5), where 2^j - 5 is prime, is a term. See A059608. - Jon E. Schoenfield, Jun 02 2019

Contains subsequence A088832.

Cf. A054024, A045768, A059608, A088834, A045770, A076496.

isok(k) = Mod(sigma(k), k) == 4; \\ Michel Marcus, Jan 04 2023

a(13) from Harvey P. Dale, Mar 20 2011
Initial term 1 inserted and a(14)-a(16) from Donovan Johnson, Mar 01 2012
Term 3 inserted by Michel Marcus, Jan 04 2023
a(18) from Jon E. Schoenfield confirmed, and a(17), a(19) added by Max Alekseyev, Jun 08 2025

A156560 Primes of the form 2^n-5.

Original entry on oeis.org

3, 11, 59, 251, 1019, 4091, 262139, 1048571, 67108859, 4294967291, 68719476731, 72057594037927931, 73786976294838206459, 332306998946228968225951765070086139, 1361129467683753853853498429727072845819, 1427247692705959881058285969449495136382746619

Offset: 1

Vincenzo Librandi, Feb 10 2009

nonn

If p = 2^n-5 is prime, then p*2^(n-1) is abundant with abundance 4 (see A088832). - Davide Rotondo, Oct 25 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..28

Corresponding n's are in A059608.

Cf. A088832.

[ a: n in [2..500] | IsPrime(a) where a is 2^n-5 ];

Select[Table[2^n-5,{n,2,400}],PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

for(n=1,300,q=2^n-5;if(isprime(q),print(q))) /* gives more terms in <10secs */ \\ Joerg Arndt, Dec 03 2010

a(n) = 2^A059608(n) - 5.

Edited by Zak Seidov

A067193 Numbers k such that sigma(k) == 4 (mod phi(k)).

Original entry on oeis.org

24, 27, 44, 66, 75, 170, 944, 1200, 16064, 260864, 4189184, 17179541504, 274876596224

Offset: 1

Benoit Cloitre, Feb 19 2002

a(14) > 10^12. 1125899822956544 and 4611686013058678784 are also terms. - Donovan Johnson, Feb 29 2012

a(14) > 10^13. If 2^j-5 is prime (A059608) and j > 3, then 2^(j-2)*(2^j-5) is a term. - Giovanni Resta, Mar 29 2020

Cf. A063514, A059608, A067192.

a(11) from Donovan Johnson, Dec 14 2009

a(12)-a(13) from Donovan Johnson, Feb 29 2012

A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

Original entry on oeis.org

12, 88, 1888, 32128, 521728, 8378368, 34359083008, 549753192448, 2251799645913088, 9223372026117357568, 2361183241263023915008, 2596148429267413634121263069790208, 2722258935367507707522529418717050175488

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

All these numbers are in A181595 because their abundance is 4, a proper divisor of m.

Cf. A059608, A156560, A181595, A181701, A000396, A181703, A045769

Rest[2^(#-1) (2^#-5)&/@(Round[N[Log[#+5]/Log[2]]]&/@Select[Table[2^t-5,{t,120}],PrimeQ])] (* Harvey P. Dale, Dec 16 2010 *)

571728 replaced with 521728 by R. J. Mathar, Dec 05 2010

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

A217348 Numbers k such that 4^k - 5 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 10, 13, 16, 18, 28, 33, 59, 65, 75, 83, 103, 113, 275, 353, 405, 568, 614, 909, 1184, 1200, 1564, 2266, 2556, 4246, 8014, 8193, 8696, 9291, 10993, 12146, 13809, 15459, 16381, 24106, 60220, 91816, 158070, 182491, 207016, 266675, 297561

Offset: 1

Vincenzo Librandi, Oct 01 2012

			28 is a term because 4^28 - 5 = 72057594037927931 is prime.

Cf. A059266, A059608, A059614.

Mathematica
```
Select[Range[10000], PrimeQ[4^# - 5] &]
```

/* Up to 620 the code produces in few seconds the first terms: */
allocatemem(10000000); for(n=2, 620, if(isprime(4^n-5), print1(n", ")));

a(n) = A059608(n+1)/2. - Daniel Starodubtsev, Mar 20 2020

a(31)-a(34) from Bruno Berselli, Oct 02 2012
a(35)-a(45) from Daniel Starodubtsev, Mar 20 2020
a(46)-a(47) derived from A059608 by Elmo R. Oliveira, Nov 28 2023

cp's OEIS Frontend

A096817 Numbers k such that 2^k - 11 is prime.

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A096819 Numbers k such that 2^k - 19 is prime.

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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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A045769 Numbers k such that sigma(k) == 4 (mod k).

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A156560 Primes of the form 2^n-5.

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A067193 Numbers k such that sigma(k) == 4 (mod phi(k)).

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A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 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.