A057247 - OEIS

%I #27 May 27 2022 21:11:05

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

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

%C The prime a(15) has 178 decimal digits. [Corrected by _Sean A. Irvine_, May 27 2022]

%H Alois P. Heinz, <a href="/A057247/b057247.txt">Table of n, a(n) for n = 1..75</a>

%H Jean-François Alcover, <a href="/A057247/a057247.txt">Table of n, a(n) for n=16..50.</a>

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

%e 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.

%p a:= proc(n) option remember; local p, m, t; p:= ithprime(n);

%p       for m do t:= 1+p*2^m; if isprime(t) then return t fi od

%p     end:

%p seq(a(n), n=1..15);  # _Alois P. Heinz_, May 27 2022

%t 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 *)

%o (Python)

%o from sympy import isprime, prime

%o def a(n):

%o     m, pn = 1, prime(n)

%o     while not isprime(1 + pn*2**m): m += 1

%o     return 1 + pn*2**m

%o print([a(n) for n in range(1, 21)]) # _Michael S. Branicky_, May 27 2022

%Y Cf. A058887, A005384, A005385, A057192.

%K nonn

%O 1,1

%A _Labos Elemer_, Jan 10 2001

%E Title clarified by _Sean A. Irvine_, May 27 2022