A056206 -id:A056206 - cp's OEIS frontend

A381041 Smallest prime p such that 3^n + p + 1 is prime.

3, 3, 3, 3, 7, 7, 3, 19, 7, 3, 3, 19, 79, 7, 7, 43, 67, 139, 127, 103, 7, 97, 3, 31, 31, 13, 379, 61, 109, 433, 3, 79, 127, 79, 67, 139, 127, 229, 7, 109, 271, 313, 3, 151, 7, 103, 67, 283, 421, 67, 43, 373, 97, 97, 97, 19, 61, 3, 157, 331, 127, 37, 139, 439, 421

Offset: 0

James S. DeArmon, Apr 20 2025

nonn

			a(0) = 3, since 3 + (3^0+1) = 5 is prime and 2 + (3^0+1) = 4 is not.
a(1) = 3, since 3 + (3^1+1) = 7 is prime and 2 + (3^1+1) = 6 is not.

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

Cf. A007051, A020483, A056206, A056208, A057673.

f:= proc(n) local t,p;
 p:= 1: t:= 3^n+1;
 do
   p:= nextprime(p);
   if isprime(p+t) then return p fi
 od;
end proc:
map(f, [$0..100]); # Robert Israel, Jun 19 2025

a[n_]:=Module[{p=2},While[!PrimeQ[p+3^n+1], p=NextPrime[p]]; p]; Array[a,65,0] (* Stefano Spezia, Apr 25 2025 *)

a(n) = my(p=2, x=3^n+1); while (!isprime(p+x), p=nextprime(p+1)); p; \\ Michel Marcus, Apr 24 2025

from sympy import isprime, nextprime
def a(n):
    p, b = 2, 3**n+1
    while not isprime(p+b):
        p = nextprime(p)
    return p
print([a(n) for n in range(65)]) # Michael S. Branicky, Apr 23 2025

from sympy import nextprime, isprime
def A381041(n):
    p = 3**n+1
    q = nextprime(p)
    while not isprime(q-p):
        q = nextprime(q)
    return q-p # Chai Wah Wu, May 01 2025

a(n) = A020483(A007051(n)). - Robert Israel, Jun 19 2025

More terms from Michael S. Branicky, Apr 23 2025

A071781 Primes p with p-2^e and p+2^e prime for some exponent e.

Original entry on oeis.org

5, 7, 11, 67, 32771

Offset: 1

Rick L. Shepherd, Jun 05 2002

For each n, p-2^e,p,p+2^e is thus an arithmetic progression of primes with difference 2^e. Note that for each n=1,2,3,4,5, only one such e exists and p-2^e=3. There are no other terms up to 20000000.

For all terms, p-2^e must, in fact, be 3 (as one of p-2^e, p and p+2^e is divisible by 3). Each corresponding arithmetic progression of primes has length 3 (p+2^(e+1) is also divisible by 3). Any additional term is too large to include here. Equivalently, this sequence is primes of the form 3+2^e such that 3+2^(e+1) is also prime; i.e., 3+2^A057732(k) is a term iff A057732(k+1) = A057732(k) + 1. Thus much more efficient than the PARI program below is to extend A057732 and examine its terms. - Rick L. Shepherd, Jun 20 2008

			67 is a term because 67 is prime and there exists e=6 such that both 67-2^6=67-64=3 and 67+2^6=67+64=131 are primes. 32771 is a term because 32771 is prime and there exists e=15 such that both 32771-2^15=32771-32768=3 and 32771+2^15=32771+32768=65539 are primes. Thus 3,67,131 and 3,32771,65539 are two sequences of primes in arithmetic progression with differences 2^6 and 2^15, respectively.

Cf. A056206, A056208.

Cf. A057732.

for(p=5,20000000,if(isprime(p),e=1; while(p-2^e>1,if(isprime(p-2^e)&&isprime(p+2^e),print1(p,","); break,e++))))

A172183 a(n) is the smallest prime of the form p^q+n, where p and q are prime, or zero if no such prime exists.

Original entry on oeis.org

5, 11, 7, 13, 13, 31, 11, 17, 13, 19, 19, 37, 17, 23, 19, 41, 8209, 43, 23, 29, 29, 31, 31, 73, 29, 53, 31, 37, 37, 79, 0, 41, 37, 43, 43, 61, 41, 47, 43, 67, 73, 67, 47, 53, 53, 71, 79, 73, 53, 59, 59, 61, 61, 79, 59, 83, 61, 67, 67, 109, 0, 71, 67, 73, 73, 191, 71, 193, 73, 79

Offset: 1

Cheng Zhang (cz1(AT)rice.edu), Jan 28 2010, Mar 02 2010

nonn

If n mod 6 = 1, both p and q must be 2, and a(n)=0 if n + 4 is not a prime. The values of a(n) for n=257,297,353,383,557 are either greater than 176 000 or 0. Several large entries: a(87) = 2^25633 + 87, a(717) = 2^3217 + 717, a(773) = 2^2539 + 773, a(927) = 2^1117 + 927.

			a(1)=5 because 5=2^2+1 is the smallest prime of the form p^q+1. a(2)=11 because 11=3^2+2. a(3)=7, because 7=2^2+3. a(17)=8209, because 8209=2^13+17. a(31)=0, because p^q+31 cannot be a prime.

Cf. A123318, A056208, A056206, A057733, A123250, A104066, A156940, A104067, A156973

For[l = {}; n = 1, n <= 70, n++, found = False; If[Mod[n, 2] == 0, For[rm = Infinity; i = 1, i < 100, i++, For[j = 1, j < 100, j++, p = Prime[i]; q = Prime[j]; r = p^q + n; If[r >= rm, Break[], If[PrimeQ[r], rm = r; found = True]]; ]; ], (* if n is odd, r=2^q+n *) If[Mod[n, 6] == 1, r = 4 + n; If[PrimeQ[r], found = True], For[j = 1, j < 1000, j++, q = Prime[j]; r = 2^q + n; If[PrimeQ[r], found = True; rm = r; Break[]]; ]; ]; ]; If[ ! found, rm = 0]; l = Append[l, rm]; ]; l

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A381041 Smallest prime p such that 3^n + p + 1 is prime.

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A071781 Primes p with p-2^e and p+2^e prime for some exponent e.

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Author

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PARI

A172183 a(n) is the smallest prime of the form p^q+n, where p and q are prime, or zero if no such prime exists.

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Author

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Examples

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Programs

Mathematica