A129962 -id:A129962 - cp's OEIS frontend

A052007 Numbers m such that 2^m + m is prime.

1, 3, 5, 9, 15, 39, 75, 81, 89, 317, 701, 735, 1311, 1881, 3201, 3225, 11795, 88071, 204129, 678561

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

Terms >= 701 are currently only strong pseudoprimes.
If m=1 (mod 6) or m=2 (mod 6) then 3 divides 2^m+m. Thus for n > 1, a(n)!=1 (mod 6) and a(n)!=2 (mod 6).
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019
Keller (see Links) notes that a Mersenne number M(2^m+m) = 2^(2^m+m) - 1 can be written as (2^m)*2^(2^m) - 1, and lists the first twelve terms of this sequence. The last known case where M(2^m+m) is prime is for m=a(4)=9, which gives the prime M(521). - Jeppe Stig Nielsen, Apr 20 2021

			2^39 + 39 = 549755813927 is prime.

W. Keller, New Cullen Primes, Math. Comp. 64 (1995), 1733-1741, S39.
Henri Lifchitz, Renaud Lifchitz, PRP Top Records. 2^n+n.

Cf. A006127, A048744, A129962.

Do[ If[ PrimeQ[ 2^n + n ], Print[ n ] ], {n, 0, 7000} ]
v={1}; Do[If[Mod[n, 2]*(Mod[n, 6]-1)!= 0&&PrimeQ[2^n+n], v=Append[v, n]; Print[v]], {n, 2, 20000}]

is(n)=isprime(2^n+n) \\ Charles R Greathouse IV, Feb 09 2017

11795 from Farideh Firoozbakht, Aug 21 2003
88071 from Hugo Pfoertner, Dec 26 2004
More terms from Henri Lifchitz submitted by Ray Chandler, Mar 02 2007

A187733 Primes of the form 2^n + n + 2.

Original entry on oeis.org

3, 5, 13, 137, 523, 524309, 134217757, 8589934627, 178405961588244985132285746181186892047843477

Offset: 1

Jaroslav Krizek, Jan 19 2013

nonn

The list of associated n is: 0, 1, 3, 7, 9, 19, 27, 33, 147, 639, ...
Primes from A052968 of the form 2^(n-1) + n + 1 for n = 1, 2, 4, 8, 10, 20, 28, 34, ...
Pairs of twin primes of forms (2^n+n; 2^n+n+2): (3; 5), (11; 13), (521; 523), ...
The prime number a(10) = 2^639 + 639 + 2 has 193 digits.
a(11) = 2^12243 + 12243 + 2 and a(12) = 2^41427 + 41427 + 2. - Giovanni Resta, Jan 23 2013
The sequence contains the subsequence 13, 137, 524309,... where the n themselves are prime, n = 3, 7, 19 (no further up to 41427). - Jaroslav Krizek, Feb 27 2013

			8589934627 is in the sequence since 8589934627 =  2^33 + 33 + 2 and 8589934627 is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..10
FactorDB, 2^n + n + 2.

Cf. A052968, A129962, A309328.

Select[Table[2^n + n + 2, {n, 60}], PrimeQ[#] &]

A100339 Primes of the form 2^q + q where q is not a prime.

Original entry on oeis.org

3, 521, 32783, 549755813927, 37778931862957161709643, 2417851639229258349412433

Offset: 1

Cino Hilliard, Jan 11 2005

nonn

The next term is 2^735+735 = 18073..35103, 222 digits long.

			For q = 9, 2^9+9 = 521 which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..11 (terms 1..7 from N. J. A. Sloane)

Cf. A006127, A057664, A081296, A100556, A129962.

Select[Table[If[!PrimeQ[n],2^n+n,0],{n,1200}],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011*)

g1(p,n)=for(x=1,n,c=composite(x);y=p^c+c;if(gcd(y,c)==1,if(isprime(y),print1 (y",")))) composite(n) = \ the n-th composite number { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

a(n) = A006127(A100556(n-1)) for n >= 2. - Amiram Eldar, Jun 30 2024

A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2.

Original entry on oeis.org

3, 7, 11, 17, 23, 31, 37, 47, 59, 67, 73, 101, 127, 131, 191, 223, 229, 239, 251, 257, 383, 401, 457, 479, 503, 521, 577, 991, 997, 1019, 1031, 1153, 1601, 1993, 2039, 2053, 2069, 3583, 3593, 3851, 3967, 4079, 4091, 4099, 4111, 4133, 6143, 6211

Offset: 1

Robin Garcia, Feb 11 2014

nonn

These prime numbers can be written in the numeral system described in A235860 (perhaps not minimally) this way : 2..21..12..2 (or 1..12..2) k twos followed to the right by l ones, followed to the right by m twos.
k can be zero, with the arbitrary limitation, when k = 0, l <= m.
When k = m = 1 we get primes of the form 2^(l + 2) + l + 2.
It must be noted these primes include the Mersenne primes 3, 7, 31, 127, 8191, ... as well as the Fermat primes 3, 5, 17, 257, 65537, with the exception of 5. Mersenne primes can be represented by a one followed to the right by an even number of twos (if the number of twos is odd, the Mersenne number is divisible by 3) with the exception of 3 represented as 12. The Fermat numbers can be represented with three ones followed to the right by a Mersenne number of twos (2^t - 1 (t = 0, 1, 2, 3, 4, 5,...)) :  3 = 111 instead of shorter 12, 5 = 1112 instead of shorter  21, 17 = 111222, 257 = 1112222222, 65537 = 111222222222222222. The composite (divisible by 641) 2^32 + 1 : three ones followed to the right by thirty one twos. The second Fermat prime: 5, appears in this sequence if we let l > m and l <= 3 when k = 0.
By A235860 3, 7 , 17 and 31 can be represented as 12, 122, 111222, 12222 cases when k=0 (primes of the form 2^(m+1) + l - 2: 31 = 2^5 +1 -2). And 11, 73, 191 as 212, 211122, 2122222 (73 = 2^7 - 2^6 + 2^3 + 3 - 2).

			For k=l=m=1, 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2 = 2^4 - 2^3 + 2^2 + 1 - 2 = 16 - 8 + 4 + 1 - 2 = 11, so 11 is in the sequence.

Cf. A235860, A236547, A007931, A129962, A237816

n=10^5;e=89;a=1;if(a%2==0,a=a+1);b=ceil(log(n)/log(2));i=0;d=floor(b^(2.5));v=vector(d);for(n=0,b,for(p=a,b,if(n==0,x=p,x=b);forstep(m=a,x,2,c=2^(n+m+p+1)-2^(m+p+1)+2^(p+1)+m-2;if(isprime(c),i++;v[i]=c))));w=vecsort(v,,8);u=vector(#(w)-1);for(j=1,#(w)-1,u[j]=w[j+1]);if(e>#(u),e=#(u));s=vector(e);for(k=1,e,s[k]=u[k];print1(s[k], ", "))

A273942 Primes of the form 3^k + k.

Original entry on oeis.org

11, 6569, 16677181699666603

Offset: 1

Vincenzo Librandi, Jun 06 2016

Terms given correspond to k in A057900.

a(4) has 731 digits.

Amiram Eldar, Table of n, a(n) for n = 1..4

Cf. A057900, A104743, A129962, A129963.

[a: n in [0..100] | IsPrime(a) where a is 3^n+n];

Select[Table[3^n + n, {n, 1, 1000}], PrimeQ]

a(n) = A104743(A057900(n)). - Amiram Eldar, Jul 27 2025

A359735 Let f(s,n) = 2^n + s*n, with s in {-1, 1}. Let c be the number of primes out of the pair f(-1,n), f(1,n). If only f(-1,n) is prime, a(n) = -1, otherwise a(n) = c.

Original entry on oeis.org

0, 1, -1, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Jean-Marc Rebert, Jan 12 2023

sign

			f(-1,1) = 2^1 - 1 = 1, is not prime and f(1,1) = 2^1 + 1 = 3 is prime, so a(1) = 1.
f(-1,2) = 2^2 - 2 = 2, is prime and f(1,2) = 2^2 + 2 = 6 = 2 * 3 is not prime, so a(2) = -1.
f(-1,3) = 2^3 - 3 = 5, is prime and f(1,3) = 2^3 + 3 = 11 is prime, so a(3) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A006127, A048744, A052007, A129962.

f(s,n)=2^n+s*n
a(n)=my(a=isprime(f(-1,n)),b=isprime(f(1,n)),c=a+b); if(c==1&&a==1,return(-1),return(c))

a(n) can be = 2 only if n = 6*m + 3 for m >= 0 and m is not congruent to {0, 4} mod 5, not congruent to {2, 4} mod 7, not congruent to {6, 7} mod 11 and not congruent to {3, 9} mod 13. Does a(n) = 2 for n > 9 exist? - Thomas Scheuerle, Jan 12 2023

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A052007 Numbers m such that 2^m + m is prime.

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A187733 Primes of the form 2^n + n + 2.

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A100339 Primes of the form 2^q + q where q is not a prime.

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A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2.

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A273942 Primes of the form 3^k + k.

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A359735 Let f(s,n) = 2^n + s*n, with s in {-1, 1}. Let c be the number of primes out of the pair f(-1,n), f(1,n). If only f(-1,n) is prime, a(n) = -1, otherwise a(n) = c.

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