A056206 -id:A056206 - cp's OEIS frontend

A056208 Primes p+2^n arising in A056206.

3, 5, 7, 11, 19, 37, 67, 131, 263, 523, 1031, 2053, 4099, 8209, 16421, 32771, 65539, 131101, 262147, 524341, 1048583, 2097169, 4194371, 8388619, 16777259, 33554473, 67108961, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Labos Elemer, Oct 06 2000

nonn

			n=13, 2^13+p=8192+p is not prime for p=2,3,5,7,11,13. At first, for p=17, 8209 is prime; Primes obtained also for many larger p, the next is 8221. So a(13)=8209, the smallest one.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A056206.

f:= proc(n) local p,q,t;
  t:= 2^n; p:= 1;
  do
    p:= nextprime(p);
    q:= p+t;
    if isprime(q) then return q fi
  od
end proc:
map(f, [$0..50]); # Robert Israel, Aug 22 2019

a[n_] := Module[{p = 1, q, t = 2^n}, While[True, p = NextPrime[p]; q = p+t; If[PrimeQ[q], Return[q]]]];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 09 2020, after Maple *)

a(0)=3 inserted by Robert Israel, Aug 22 2019

A057663 Primes p such that p + 2^p is also a prime.

Original entry on oeis.org

3, 5, 89, 317, 701

Offset: 1

Labos Elemer, Oct 16 2000

Different from A056206, where, e.g., at n=89, 89 is not minimal, A056206(89)=29 and not 89.
a(6) > 27479. - Ralf Stephan, Oct 23 2002
Intersection of A000040 and A052007. - Iain Fox, Nov 08 2017
a(6) > 678561. - Iain Fox, Nov 08 2017
Every term other than 3 is congruent to 5 (mod 6). - Arkadiusz Wesolowski, Nov 14 2017
These terms satisfy phi(k + 2^k) = phi(k) + 2^k, where phi is A000010, the Euler totient function. Conjecture: this sequence gives all numbers k that satisfy the condition phi(k + 2^k) = phi(k) + 2^k. - Juri-Stepan Gerasimov, May 23 2019

			q=3, 2^3 + 3 = 11 a prime.

Cf. A000010, A006127, A056206, A056208, A057664, A057665.

[p: p in PrimesUpTo(1000) | IsPrime(2^p+p) ] // Vincenzo Librandi, Aug 07 2010

Select[Prime@ Range[10^3], PrimeQ[# + 2^#] &] (* Michael De Vlieger, Nov 08 2017 *)

lista(nn) = forprime(p=3, nn, if(ispseudoprime(p + 2^p), print1(p, ", "))) \\ Iain Fox, Nov 13 2017

[n for n in (1..1000) if is_prime(n) and is_prime(2^n+n)] # G. C. Greubel, May 24 2019

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

A057673 Smallest prime p such that |2^n - p| is a prime.

Original entry on oeis.org

3, 5, 2, 3, 3, 3, 3, 19, 5, 3, 3, 19, 3, 13, 3, 19, 17, 13, 5, 19, 3, 19, 3, 37, 3, 61, 5, 79, 89, 3, 41, 19, 5, 79, 41, 31, 5, 31, 107, 7, 167, 31, 11, 67, 17, 139, 167, 127, 59, 139, 71, 139, 47, 379, 53, 67, 5, 13, 137, 607, 107, 31, 167, 409, 59, 79, 5, 19, 23, 19, 71, 577, 107

Offset: 0

Labos Elemer, Oct 19 2000

nonn

The absolute value is relevant only for first two terms, 2^0-a(0) = 1-3 = -2, 2^1-a(1) = 2-5 = -3. According to Goldbach's conjecture, every even number > 2 is the sum of two primes, which implies that for all further terms, a(n) < 2^n. - M. F. Hasler, Jan 13 2011

			n=7, 2^n=128. The smallest terms subtracted from 128 resulting in a prime are 1,15,19,... Neither 1 nor 15 are primes but 19 is a prime. It gives 109=128-19, so a(n)=19.

Hugo Pfoertner, Table of n, a(n) for n = 0..5000

Analog of A056206. Cf. A056208, A057662.

f[n_] := Block[{p = 2}, While[! PrimeQ[2^n - p], p = NextPrime@ p]; p]; Array[f, 60, 0]

A057673(n)=forprime( p=1,default(primelimit), ispseudoprime(abs(2^n-p))& return(p))

Offset corrected and initial term added by M. F. Hasler, Jan 13 2011

A057662 Smallest prime q such that q + 2^prime(n) is a prime, where prime(n) is the n-th prime.

Original entry on oeis.org

3, 3, 5, 3, 5, 17, 29, 53, 11, 11, 11, 29, 71, 29, 5, 5, 131, 197, 3, 11, 29, 23, 89, 29, 359, 149, 239, 239, 881, 281, 29, 3449, 197, 683, 389, 683, 101, 1283, 83, 191, 1181, 197, 5, 71, 107, 101, 71, 4001, 1433, 251, 431, 29, 1361, 89, 509, 83, 2459, 941, 101, 197

Offset: 1

Labos Elemer, Oct 16 2000

nonn

			For n = 6, prime(6) = 13, 8192 + 17 = 8209 is a prime but 8192 + 2, 8182 + 3, ..., 8192 + 13 are not, so a(6) = 17.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A056206, A056208.

a[n_] := Module[{m = 2^Prime[n], q = 2}, While[!PrimeQ[q + m], q = NextPrime[q]]; q]; Array[a, 60] (* Amiram Eldar, Mar 13 2025 *)

a(n) = my(P=2^prime(n), q=2); while (!isprime(P+q), q = nextprime(q+1)); q; \\ Michel Marcus, Mar 04 2022

a(n) = A056206(A000040(n)).

A361679 A(n,k) is the n-th prime p such that p + 2^k is also prime; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 3, 5, 3, 7, 11, 3, 5, 13, 17, 5, 7, 11, 19, 29, 3, 11, 13, 23, 37, 41, 3, 7, 29, 31, 29, 43, 59, 7, 11, 19, 41, 37, 53, 67, 71, 11, 13, 23, 37, 47, 43, 59, 79, 101, 7, 29, 37, 29, 43, 71, 67, 71, 97, 107, 5, 37, 59, 61, 53, 67, 107, 73, 89, 103, 137

Offset: 1

Alois P. Heinz, Mar 20 2023

			Square array A(n,k) begins:
    3,   3,   3,   3,   5,   3,   3,   7,  11,   7, ...
    5,   7,   5,   7,  11,   7,  11,  13,  29,  37, ...
   11,  13,  11,  13,  29,  19,  23,  37,  59,  67, ...
   17,  19,  23,  31,  41,  37,  29,  61,  89,  73, ...
   29,  37,  29,  37,  47,  43,  53,  97, 101,  79, ...
   41,  43,  53,  43,  71,  67,  71, 103, 107, 127, ...
   59,  67,  59,  67, 107,  73,  83, 127, 131, 139, ...
   71,  79,  71,  73, 131, 103, 101, 163, 149, 157, ...
  101,  97,  89,  97, 149, 109, 113, 193, 179, 163, ...
  107, 103, 101, 151, 167, 127, 149, 211, 197, 193, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Columns k=1-10 give: A001359, A023200, A023202, A049488, A049489, A049490, A049491, A361483, A361484, A361485.
Row n=1 gives A056206.
Main diagonal gives A361680.
Cf. A000040.

A:= proc() option remember; local f; f:= proc() [] end;
      proc(n, k) option remember; local p;
        p:= `if`(nops(f(k))=0, 1, f(k)[-1]);
        while nops(f(k))

A127003 Smallest prime p such that p+10^n is also prime.

Original entry on oeis.org

2, 3, 3, 13, 7, 3, 3, 19, 7, 7, 19, 3, 61, 37, 31, 37, 61, 3, 3, 97, 151, 193, 139, 157, 7, 13, 67, 103, 331, 379, 211, 271, 709, 61, 193, 103, 67, 43, 547, 3, 139, 109, 991, 79, 31, 439, 937, 193, 193, 2089, 151, 457, 919, 223, 31, 157, 3, 607, 601, 19, 7, 379, 991, 307

Offset: 0

Rick L. Shepherd, Jan 01 2007

nonn

			a(3)=13 because 13 is the smallest prime p such that p+10^3 (=1013) is also prime: For the primes < 13, 1002=2*3*167, 1003=17*59, 1005=3*5*67, 1007=19*53 and 1011=3*337, respectively, are not prime.

Pierre CAMI, Table of n, a(n) for n = 0..1000

Cf. A127004, A056206.

s={};Do[p=0;Until[PrimeQ[Prime[p]+10^n],p++];AppendTo[s,Prime[p]],{n,0,63}];s (* James C. McMahon, Dec 29 2024 *)

A057674 Primes -p+2^n with smallest p prime, arising in A057674.

Original entry on oeis.org

-3, 2, 5, 13, 29, 61, 109, 251, 509, 1021, 2029, 4093, 8179, 16381, 32749, 65519, 131059, 262139, 524269, 1048573, 2097133, 4194301, 8388571, 16777213, 33554371, 67108859, 134217649, 268435367, 536870909, 1073741783, 2147483629, 4294967291, 8589934513, 17179869143

Offset: 0

Labos Elemer, Oct 19 2000

sign

			n=1, 2^1=2. If 2,3,5 are subtracted from 2, then 0,-1 and -3 arise, of which -3 is a prime so a(1)=-3. n=11, 2048-p=q. At first p=29 gives the prime q=2029.

Cf. A056206, A056208, A056662, A057674.

A057676 Smallest prime q such that 2^prime(n) - q is prime.

Original entry on oeis.org

2, 3, 3, 19, 19, 13, 13, 19, 37, 3, 19, 31, 31, 67, 127, 379, 607, 31, 19, 577, 181, 67, 97, 31, 349, 619, 97, 919, 31, 211, 577, 181, 13, 397, 31, 829, 19, 577, 577, 103, 1669, 199, 19, 31, 439, 1021, 601, 1621, 2017, 733, 3, 199, 2113, 619, 1861, 1297, 241, 967

Offset: 1

Labos Elemer, Oct 19 2000

nonn

			For n = 4, prime(4) = 11, 2^11 = 2048, p2 = 2048-p1 is satisfied at first with prime p1 = 19 resulting in prime p2 = 2029, so a(4) = 19.
For n = 31, prime(31) = 127, p2 = 2^127-p1 is satisfied first with p1 = 577 and p2 = 170141183460469231731687303715884105151, so a(31) = 577.

Amiram Eldar, Table of n, a(n) for n = 1..560

Cf. A057662, A056206, A056208.

spq[n_]:=Module[{p=2,t=2^Prime[n]},While[!PrimeQ[t-p],p=NextPrime[p]];p]; Array[spq,60] (* Harvey P. Dale, Jul 13 2025 *)

a(n) = {my(p = 1 << prime(n), q = 2); while(!isprime(p - q), q = nextprime(q + 1)); q;} \\ Amiram Eldar, Feb 18 2025

Offset corrected by Amiram Eldar, Feb 18 2025

A371065 a(1)=2; for n > 1, a(n) is the least prime number p > a(n-1) such that p + 2^(n-1) is a prime number.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 37, 53, 61, 89, 127, 131, 157, 197, 223, 269, 307, 359, 367, 419, 463, 491, 547, 593, 607, 641, 643, 701, 823, 947, 1213, 1229, 1237, 1319, 1327, 1451, 1723, 2381, 3019, 3299, 3307, 3371, 3847, 4493, 4621, 4931, 5179, 5783, 6043, 6197, 6469

Offset: 1

Ahmad J. Masad, Mar 09 2024

nonn

			For n=5, the preceding term a(4)=11 and 2^(5-1)=16, so a(5) is the least prime p > 11 such that p+16 is a prime too, which is p = 13 = a(5).
From _Michael De Vlieger_, Mar 10 2024: (Start)
Table of first terms:
   n   a(n)  2^(n+1)  a(n)+2^(n+1)
  -------------------------------
   1      2       1         3
   2      3       2         5
   3      7       4        11
   4     11       8        19
   5     13      16        29
   6     29      32        61
   7     37      64       101
   8     53     128       181
   9     61     256       317
  10     89     512       601
  11    127    1024      1151
  12    131    2048      2179
  ... (End)

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A108184, A056206.

a[1] = 2; a[n_] := a[n] = Module[{p = NextPrime[a[n - 1]]}, While[! PrimeQ[p + 2^(n - 1)], p = NextPrime[p]]; p]; Array[a, 50] (* Amiram Eldar, Mar 10 2024 *)

cp's OEIS Frontend

A056208 Primes p+2^n arising in A056206.

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A057663 Primes p such that p + 2^p is also a prime.

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

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A057673 Smallest prime p such that |2^n - p| is a prime.

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A057662 Smallest prime q such that q + 2^prime(n) is a prime, where prime(n) is the n-th prime.

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A361679 A(n,k) is the n-th prime p such that p + 2^k is also prime; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A127003 Smallest prime p such that p+10^n is also prime.

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Mathematica

A057674 Primes -p+2^n with smallest p prime, arising in A057674.

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