A242944 -id:A242944 - cp's OEIS frontend

A078686 Primes prime(k) such that 2^k - prime(k) is also prime.

5, 31, 67, 89, 101, 283, 503, 1039, 1129, 2069, 3457, 5641, 45763, 71483, 86599, 112921, 161411, 210173, 211007, 232741, 245269, 479029, 1078589

Offset: 1

Benoit Cloitre, Dec 17 2002

The original definition ("Primes p such that the minimum value of |p-2^x|, x>0, is also a prime") produces A188677, not the terms shown here. - N. J. A. Sloane, Apr 01 2011.

The mystery of this definition was solved by Robert G. Wilson v, Jul 06 2014, who also remarks that if instead we ask for odd primes, and therefore the index is one less than that for all primes, the sequence would begin: 11, 13, 17, 19, 23, 37, 61, 233, 257, 1553, 2879, 4919, 6389, 7621, 8081, 35593, 37951, 96263, 206419, ..., . If we count 1 amongst the primes (A008578), then the sequence would begin: 2, 3, 5, 11, 17, 167, 193, 197, 433, 4111, 9173, 42929, 95279, 98897, 139409, 142567, 228617, ..., .

			a(1)=5 since 5 is the third prime number and 2^3-5 = 3 is prime. - _Robert G. Wilson v_, Jul 06 2014

Corresponding k are in A078583.

Cf. A000040, A242944, A188677.

p = 2; lst = {}; While[p < 760001, If[ PrimeQ[ 2^PrimePi@ p - p], AppendTo[lst, p]; Print@ p]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jul 06 2014 *)
Select[Table[{n,Prime[n]},{n,40000}],PrimeQ[2^#[[1]]-#[[2]]]&][[All,2]] (* Harvey P. Dale, Feb 19 2020 *)

Edited (corrected title) and extended by Robert G. Wilson v, Jul 06 2014

a(23) from Michael S. Branicky, May 29 2025

A077375 Numbers k such that 2^k + prime(k) is prime.

Original entry on oeis.org

2, 3, 4, 5, 12, 13, 14, 23, 57, 106, 226, 227, 311, 373, 1046, 1298, 1787, 1952, 2130, 2285, 2670, 3254, 3642, 4369, 13559, 33418, 66830, 213810

Offset: 1

Jason Earls, Nov 30 2002

All terms < 5000 correspond to certified primes (Primo 2.2.0 beta). - Ryan Propper, Aug 08 2005

			2^12 + prime(12) = 4096 + 43 = 4139 is a prime, so 12 is a term.

H. Lifchitz, Mersenne and Fermat primes field

Corresponding prime(k) are in A242944.

Cf. A078583, A078686.

a(25)-a(26) from Ryan Propper, Aug 08 2005
a(27) from Henri Lifchitz added by Jason Earls, Jan 12 2008
a(28) from Michael S. Branicky, Jun 01 2025

A227126 Primes prime(k) such that 2^(k+1) - prime(k) is also prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 167, 193, 197, 433, 4111, 9173, 42929, 95279, 98897, 139409, 142567, 228617, 329333, 344209, 791191, 829177, 1274509, 1284037, 2432791, 2443741

Offset: 1

Gerasimov Sergey, Jul 02 2013

The corresponding primes 2^(k + 1) - prime(k) are 2, 5, 11, 53, 239, 1099511627609, 35184372088639, ...

The prime indices k are 1, 2, 3, 5, 7, 39, 44, 45, 84, 566, 1137, ...

			5 is a term because 5 is the 3rd prime, and 2^(3+1) - 5 = 16 - 5 = 11 which is a prime
11 is in the sequence because 11 = prime(5) and 2^(5 + 1) - 11 = 64 - 11 = 53 is a prime.

Cf. A078686, A244913, A244916, A242944, A228021.

p = 2; lst = {}; While[p < 850001, If[ PrimeQ[ 2^(PrimePi@ p +1) - p], AppendTo[lst, p]; Print@ p]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jul 09 2014 *)

lista(nn) = {ip = 1; forprime(p=2, nn, if (isprime(2^(ip+1)-p), print1(p, ", ")); ip++;);} \\ Michel Marcus, Jul 12 2014

a(3), a(6), a(8)-a(12) from Joerg Arndt, Jul 03 2013
Corrected and extended through a(21) by Robert G. Wilson v, Jul 09 2014
Entry revised by N. J. A. Sloane, Jan 02 2019, incorporating data from a later submission from Robert G. Wilson v
a(22)-a(25) from Michael S. Branicky, May 31 2025

A228021 Prime(k) such that 2^(k - 1) + prime(k) is also prime.

Original entry on oeis.org

2, 3, 29, 89, 251, 659, 937, 1307, 1453, 8179, 9391, 12097, 28499, 83969, 101209, 120739, 730993

Offset: 1

Juri-Stepan Gerasimov, Aug 03 2013

The primes indices k are 1, 2, 10, 24, 54, 120, 159, 214, 231, 1027, 1161, 1447, 3100, 8188, 9695, 11363 ...
The corresponding primes 2^(k - 1) + prime(k) are 3, 5, 541, 8388697,...
a(18) > 2*10^6. - Michael S. Branicky, Apr 16 2025

			29 is in the sequence because 29 = prime(10) and 2^(10 - 1) + 29 = 512 + 29 = 541 is prime.

Cf. A077375, A227126, A242944, A244913, A244916.

for i from 1 do
    p := ithprime(i) ;
    if isprime(p+2^(i-1)) then
       printf("%d,\n",p) ;
    end if;
end do: # R. J. Mathar, Jul 12 2014

p = 2; lst = {}; While[p < 730001, If[ PrimeQ[ 2^(PrimePi@ p-1) + p], AppendTo[lst, p]; Print@ p]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jul 09 2014 *)

lista(nn) = {ip = 1; forprime(p=2, nn, if (isprime(2^(ip-1)+p), print1(p, ", ")); ip++;);} \\ Michel Marcus, Jul 12 2014

a(3) - a(9) from _Olivier Gérard_, Aug 01 2013
a(10) - a(15) from Robert G. Wilson v, Aug 01 2013
a(16) from Robert G. Wilson v, Jul 09 2014
a(17) from Michael S. Branicky, Apr 14 2025

A244916 Primes prime(k) such that 2^(k+1) + prime(k) is also prime.

Original entry on oeis.org

3, 31, 71, 97, 107, 277, 307, 641, 907, 967, 1009, 1447, 3463, 3527, 7757, 8167, 250867, 279047, 1107791, 1176671, 1538399, 1594909, 2450017

Offset: 1

Robert G. Wilson v, Jul 09 2014

Cf. A078686, A242944, A227126, A228021.

for i from 1 do
        p := ithprime(i) ;
        if isprime(p+2^(i+1)) then
                printf("%d,\n",p) ;
        end if;
end do: # R. J. Mathar, Jul 12 2014

p = 2; lst = {}; While[p < 900000, If[ PrimeQ[ 2^(PrimePi@ p +1) + p], AppendTo[lst, p]; Print@ p]; p = NextPrime@ p]; lst

lista(nn) = {ip = 1; forprime(p=2, nn, if (isprime(2^(ip+1)+p), print1(p, ", ")); ip++;);} \\ Michel Marcus, Jul 12 2014

a(19)-a(23) from Michael S. Branicky, May 31 2025

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A078686 Primes prime(k) such that 2^k - prime(k) is also prime.

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A077375 Numbers k such that 2^k + prime(k) is prime.

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A227126 Primes prime(k) such that 2^(k+1) - prime(k) is also prime.

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A228021 Prime(k) such that 2^(k - 1) + prime(k) is also prime.

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A244916 Primes prime(k) such that 2^(k+1) + prime(k) is also prime.

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Maple

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