A181697 Length of the complete Cunningham chain of the first kind starting with prime(n).

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, 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, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1

Offset

1,1

Comments

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

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

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.

Wikipedia, Cunningham chain

Formula

a(n) < prime(n) for n > 1; see Löh (1989), p. 751. - Jonathan Sondow, Oct 28 2015

max(a(n), A181715(n)) = A263879(n) for n > 2. - Jonathan Sondow, Oct 30 2015

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A005602, A181715, A263879.

See also A075712, A338945.

Sequence in context: A187059 A267120 A267484 * A317175 A257480 A181696

Adjacent sequences: A181694 A181695 A181696 * A181698 A181699 A181700

Keyword

nonn

Author

M. F. Hasler, Nov 17 2010

Extensions

Definition clarified by Jonathan Sondow, Oct 28 2015

Status

approved