A078686 -id:A078686 - cp's OEIS frontend

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

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

A078583 Numbers k such that 2^k - prime(k) is prime.

Original entry on oeis.org

3, 11, 19, 24, 26, 61, 96, 175, 189, 312, 483, 741, 4741, 7082, 8421, 10695, 14802, 18824, 18892, 20655, 21653, 39937, 84160

Offset: 1

Robert G. Wilson v, Nov 30 2002

a(23) is greater than 50000 - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Dec 05 2004

a(24) is greater than 200000. - Michael S. Branicky, Jan 03 2025

Corresponding prime(k) are in A078686.

Cf. A077375.

Do[ If[ PrimeQ[2^n - Prime[n]], Print[n]], {n, 1, 10^5}]

a(14)-a(22) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Dec 05 2004

a(23) from Michael S. Branicky, Dec 31 2024

A242944 Primes prime(k) such that 2^k + prime(k) is also prime.

Original entry on oeis.org

3, 5, 7, 11, 37, 41, 43, 83, 269, 577, 1429, 1433, 2063, 2549, 8353, 10639, 15299, 16927, 18637, 20201, 24007, 30097, 34039, 41777, 146609, 394249, 839203, 2955319

Offset: 1

Robert G. Wilson v, Jun 20 2014

If instead we ask for odd primes, and therefore the index is one less than that for all primes, the sequence would begin: 3, 29, 89, 251, 659, 937, 1307, 1453, 8179, 9391, 12097, 28499, 83969, 101209, 120739, ..., .

If we count 1 amongst the primes (A008578), then the sequence would begin: 1, 3, 31, 71, 97, 107, 277, 307, 641, 907, 967, 1009, 1447, 3463, 3527, 7757, 8167, ..., .

Corresponding k are in A077375.

Cf. A078583, A078686, A056206, A243428.

p = 2; lst = {}; While[p < 760001, If[ PrimeQ[p + 2^PrimePi@ p], AppendTo[ lst, p]; Print@ p]; p = NextPrime@ p; c++]; lst
Select[Table[{n,Prime[n]},{n,3000}],PrimeQ[#[[2]]+2^#[[1]]]&][[;;,2]] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Mar 04 2024 *)

a(27) from Michael S. Branicky, May 29 2025 using A077375.

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

A244913 Primes prime(k) such that 2^(k-1) - prime(k) is also prime.

Original entry on oeis.org

11, 13, 17, 19, 23, 37, 61, 233, 257, 1553, 2879, 4919, 6389, 7621, 8081, 35593, 37951, 96263, 206419, 596803, 1202837, 2837851

Offset: 1

Robert G. Wilson v, Jul 09 2014

a(22) > 1211303. - J.W.L. (Jan) Eerland, Dec 08 2022

a(23) > 3000000. - Michael S. Branicky, Jun 03 2025

Cf. A078686, A227126.

for i from 1 do
    p := ithprime(i) ;
    if isprime(2^(numtheory[pi](p-1))-p) then
        printf("%d,\n",p) ;
    end if;
end do: # R. J. Mathar, Jul 11 2014

p = 2; lst = {}; While[p < 800001, If[ PrimeQ[ 2^(PrimePi@ p-1) - p], AppendTo[lst, p]; Print@ p]; p = NextPrime@ p]; lst
n=1;Monitor[Parallelize[While[True,If[PrimeQ[2^(PrimePi[Prime[n]-1])-Prime[n]],Print[Prime[n]]];n++];n],n] (* J.W.L. (Jan) Eerland, Dec 08 2022 *)

is(n)=isprime(n) && isprime(2^primepi(n-1)-n) \\ Charles R Greathouse IV, Feb 25 2017

{p in A000040: 2^[A000720(p-1)]-p in A000040}. - R. J. Mathar, Jul 11 2014

a(21) from J.W.L. (Jan) Eerland, Dec 08 2022

a(22) from Michael S. Branicky, Jun 02 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

A188677 Primes p such that the minimum value of |p-2^x|, x>0, is also a prime.

Original entry on oeis.org

11, 13, 19, 23, 29, 37, 43, 53, 59, 61, 67, 71, 83, 97, 109, 131, 139, 151, 157, 181, 197, 227, 233, 239, 251, 263, 269, 293, 317, 353, 359, 383, 409, 433, 439, 499, 509, 523, 541, 571, 601, 613, 619, 643, 661, 691, 709, 739, 751, 773, 797, 827, 857

Offset: 1

Benoit Cloitre & Robert G. Wilson v, Apr 08 2011

nonn

Originally submitted by Benoit Cloitre, Dec 17 2002 as A078686 and corrected by Robert G. Wilson v, Apr 08 2011.

Cf. A078686, A086081, A091932.

fQ[n_] := Block[{x = Floor@ Log2@ n}, PrimeQ@ Min[n - 2^x, 2^(x+1) - n]]; Select[ Prime@ Range@ 150, fQ] (* Robert G. Wilson v, Apr 08 2011 *)

is(n)=if(isprime(n),my(x=log(n)\log(2));isprime(min(abs(n-1<Charles R Greathouse IV, Jan 10 2013

Intersection of A086081 and A091932. - Robert G. Wilson v, May 27 2011

A244917 Smallest prime p such that p - 2^e is also prime in exactly n cases for nonnegative integers e.

Original entry on oeis.org

2, 3, 7, 19, 139, 829, 3331, 32941, 176417, 854929, 2233531, 12699571, 47924959, 763597201, 5775760189

Offset: 0

Robert G. Wilson v, Jul 09 2014

The exponent e is obviously limited to 0 <= e <= log_2(p).
The sequence is obtained by building a greedy prime index inverse of A078687, which is 1, 2, 4, 8, 34, ..., followed by lookup in the primes, A000040.
From Robert G. Wilson v, Sep 12 2014: (Start)
2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, …, ;
3, 5, 17, 29, 41, 53, 59, 89, 97, 137, 163, 179, 191, …, ;
7, 11, 13, 23, 31, 37, 43, 47, 67, 71, 73, 79, 101, …, ;
19, 61, 83, 131, 167, 227, 229, 241, 271, 293, 353, …, ;
139, 181, 199, 571, 601, 619, 677, 691, 1217, 1231, …, ;
829, 1487, 2131, 2341, 2551, 2971, 4051, 4261, 4583, …, ;
3331, 12109, 14551, 17393, 22279, 22307, 22741, …, ;
32941, 34369, 44029, 49433, 67189, 95717, 99833, …, ;
176417, 304771, 314723, 314779, 349667, 414707, …, ;
854929, 1297651, 1328927, 1784723, 2164433, 2488909, …, ;
2233531, 6026089, 7475389, 7623229, 9644911, …, ;
12699571, 18464123, 52849879, 78127339, 79303579, …, ;
47924959, 153309649, 204797059, 248685923, 273865219, …, ;
763597201, 1194032507, 1522018741, 1833343669, …, ;.
(End)

			a(3) = 19 since 19-2^1=17, 19-2^3=11 & 19-2^4=3 and there exists no prime less than 19 which exhibits this characteristic.

Cf. A078686.

f[n_] := Length@ Table[q = p - 2^exp; If[ PrimeQ@ q, {q}, Sequence @@ {}], {exp, 0, Floor@ Log2@ p}]; t = Table[0, {20}]; p = 2; While[p < 100000001, a = f@ p; If[ t[[a]] == 0, t[[a]] = p; Print[{a, p}]]; p = NextPrime@ p]; t

a(14) from Robert G. Wilson v, Sep 12 2014

cp's OEIS Frontend

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

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A078583 Numbers k such that 2^k - prime(k) is prime.

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A242944 Primes prime(k) such that 2^k + prime(k) is also prime.

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

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A244913 Primes 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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A188677 Primes p such that the minimum value of |p-2^x|, x>0, is also a prime.

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A244917 Smallest prime p such that p - 2^e is also prime in exactly n cases for nonnegative integers e.

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