A226178 Exponents n such that 2^n - previous_prime(2^n) = next_prime(2^n) - 2^n.

2, 6, 12, 76, 181, 1099, 1820, 9229

Offset: 1

Author

Jean-François Alcover, May 30 2013

Keywords

• nonn
• hard
• more

Comments

The differences next_prime(2^n) - 2^n are respectively: 1, 3, 3, 15, 165, 1035, 663, 2211.

If it exists, a(9) > 10000. - Hugo Pfoertner, Feb 06 2021

Examples

			2^6 = 64, next prime = 67, previous prime = 61, 67-64 = 64-61 = 3, hence 6 is in the sequence.
		

Crossrefs

Cf. A000079, A014210, A014234, A117387, A145025.

Cf. A013597, A013603, A340707.

Programs

Mathematica

Reap[Do[m = 2^n; p = NextPrime[m, -1]; q = NextPrime[m]; If[p + q == 2*m, Print[n]; Sow[n]], {n, 2, 10^4}]][[2, 1]]

PARI

isok(n) = my(p=2^n); p-precprime(p-1) == nextprime(p+1) - p; \\ Michel Marcus, Oct 02 2019

PARI

for(n=2,1100,my(p2=2^n,pn=nextprime(p2),pp=p2-pn+p2);if(ispseudoprime(pp),if(precprime(p2)==pp,print1(n,", ")))) \\ Hugo Pfoertner, Feb 06 2021

Python

from itertools import count, islice
from sympy import isprime, nextprime
def A226178_gen(): # generator of terms
    return filter(lambda n:isprime(r:=((k:=1<A226178_list = list(islice(A226178_gen(),5)) # Chai Wah Wu, Aug 08 2022

Formula

A340707(a(n)) = 0. - Hugo Pfoertner, Feb 06 2021

Extensions

Offset 1 from Michel Marcus, Oct 02 2019

a(8) from Hugo Pfoertner, Feb 05 2021