A181697 - OEIS

%I #33 Nov 24 2021 09:02:58

%S 5,2,4,1,3,1,1,1,2,2,1,1,3,1,1,2,1,1,1,1,1,1,2,6,1,1,1,1,1,2,1,2,1,1,

%T 1,1,1,1,1,2,5,1,2,1,1,1,1,1,1,1,2,2,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,

%U 1,1,1,4,1,1,1,1,1,1,1,1,2,1,2,1,1,2,1,1

%N Length of the complete Cunningham chain of the first kind starting with prime(n).

%C Number of iterations x->2x+1 needed to get a composite number, when starting with prime(n).

%C prime(n) is in A005384, i.e., a Sophie Germain prime, iff a(n)>1.

%C a(n) is the least k such that 2^k * (prime(n)+1) - 1 is composite. Note that a(n) is well defined since 2^(p-1) * (p+1) - 1 is divisible by p for odd primes p. - _Jianing Song_, Nov 24 2021

%H T. D. Noe, <a href="/A181697/b181697.txt">Table of n, a(n) for n = 1..10000</a>

%H G. Löh, <a href="http://www.jstor.org/stable/2008735">Long chains of nearly doubled primes</a>, Math. Comp., 53 (1989), 751-759.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>

%F a(n) < prime(n) for n > 1; see Löh (1989), p. 751. - _Jonathan Sondow_, Oct 28 2015

%F max(a(n), A181715(n)) = A263879(n) for n > 2. - _Jonathan Sondow_, Oct 30 2015

%e 2 -> 5 -> 11 -> 23 -> 47 -> 95 = 5*19, so a(1) = 5, a(3) = 4, a(5) = 3, a(9) = 2, and a(15) = 1. - _Jonathan Sondow_, Oct 30 2015

%t Table[p = Prime[n]; cnt = 1; While[p = 2*p + 1; PrimeQ[p], cnt++]; cnt, {n, 100}] (* _T. D. Noe_, Jul 12 2012 *)

%o (PARI) a(n)= n=prime(n); for(c=1,1e9, is/*pseudo*/prime(n=2*n+1) || return(c))

%Y Cf. A005602, A181715, A263879.

%Y See also A075712, A338945.

%K nonn

%O 1,1

%A _M. F. Hasler_, Nov 17 2010

%E Definition clarified by _Jonathan Sondow_, Oct 28 2015