A340117 - OEIS

%I #11 Dec 29 2020 02:55:06

%S 2,7,5,3,53,139,157,61,1151,3833,6653,7159,30713,4093,204797,311293,

%T 360439,2555897,3014653,786431,11010037,5242877,73400311,138412031,

%U 461373431,1124073463,436207613,3288334303,10066329587,1879048183,8053063661,102005473259,40802189303,193273528303,403726925821

%N a(n) is the least prime p such that the 2-adic valuation of p+q is n, where q is the next prime after p, or 0 if there is no such p.

%C a(n)=A340116(n+1) for all n >= 2.

%C Dickson's conjecture implies that a(n) always exists, as for any n there will be k such that p = 2^(n-1)-1+k*2^n and q = p+2 = 2^(n-1)+1+k*2^n are primes.

%e a(4) = 53 because 53 is prime, the next prime is 59, 53+59 = 112 = 2^4*7, and this is the first prime p in which 2^4 appears in the factorization of p+q.

%p g:= proc(m) local k,p;

%p   for k from 2^(m-1) by 2^m do

%p     p:= prevprime(k);

%p     if nextprime(p) = 2*k-p then return p fi

%p   od

%p end proc:

%p g(0):= 2: g(1):= 7: g(2):= 5:map(g, [$0..30]);

%Y Cf. A007814, A340116.

%K nonn

%O 0,1

%A _J. M. Bergot_ and _Robert Israel_, Dec 28 2020