A057247 a(n) is the smallest prime of the form 1 + prime(n)*2^m, with m > 0.

5, 7, 11, 29, 23, 53, 137, 1217, 47, 59, 7937, 149, 83, 173

Offset: 1

Author

Labos Elemer, Jan 10 2001

Keywords

• nonn

Comments

The prime a(15) has 178 decimal digits. [Corrected by Sean A. Irvine, May 27 2022]

Examples

			Sophie-Germain primes are here at n = 1, 2, 3, 5, 9, 10, .. etc. At n = 11, p(11) = 31 and in the sequence of q = 1+31*{2, 4, 8, 16, 32, 64, 128, 256} = {63, 125, 249, 497, 993, 1985, 3969, 7937}, the first prime is 7937, so b(11) = 8, a(11) = 7937.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..75
• Jean-François Alcover, Table of n, a(n) for n=16..50.

Crossrefs

Cf. A058887, A005384, A005385, A057192.

Programs

Maple

a:= proc(n) option remember; local p, m, t; p:= ithprime(n);
      for m do t:= 1+p*2^m; if isprime(t) then return t fi od
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, May 27 2022

Mathematica

a[n_] := (For[pn = Prime[n]; p = 2, p < 3*10^8 (* large enough to compute 50 terms except a(15) *), p = NextPrime[p], m = Log[2, (p-1)/pn]; If[m > 0 && IntegerQ[m], Print["a(", n, ") = ", p]; Return[p]]]; Print["a(", n, ") not found ", p]; 0); Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 08 2016 *)

Python

from sympy import isprime, prime
def a(n):
    m, pn = 1, prime(n)
    while not isprime(1 + pn*2**m): m += 1
    return 1 + pn*2**m
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, May 27 2022

Formula

a(n) = Min{q|q is prime, p(n) is the n-th prime and q = 1+p(n)*2^b(n)}.

Extensions

Title clarified by Sean A. Irvine, May 27 2022