|
|
A181715
|
|
Length of the complete Cunningham chain of the second kind starting with prime(n).
|
|
8
|
|
|
3, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Number of iterations x -> 2x-1 needed to get a composite number, when starting with prime(n).
Dickson's conjecture implies that, for every positive integer r, there exist infinitely many n such that a(n) = r. - Lorenzo Sauras Altuzarra, Feb 12 2021
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
|
|
|
FORMULA
|
a(n) < prime(n) for n > 1; see Löh (1989), p. 751. - Jonathan Sondow, Oct 28 2015
|
|
EXAMPLE
|
2 -> 3 -> 5 -> 9 = 3^2, so a(1) = 3 and a(2) = 2. - Jonathan Sondow, Oct 30 2015
|
|
MAPLE
|
a := proc(n)
local c, l:
c, l := 0, ithprime(n):
while isprime(l) do c, l := c+1, 2*l-1: od:
c:
|
|
MATHEMATICA
|
Table[p = Prime[n]; cnt = 1; While[p = 2*p - 1; PrimeQ[p], cnt++]; cnt, {n, 100}] (* T. D. Noe, Jul 12 2012 *)
Table[-1 + Length@ NestWhileList[2 # - 1 &, Prime@ n, PrimeQ@ # &], {n, 98}] (* Michael De Vlieger, Apr 26 2017 *)
|
|
PROG
|
(PARI) a(n)= n=prime(n); for(c=1, 1e9, is/*pseudo*/prime(n=2*n-1) || return(c))
|
|
CROSSREFS
|
Cf. A000040, A005382, A005408, A005602, A005603, A181697, A263879, A285700, A285706, A057326, A057327, A057328, A057329, A057330, A064812.
Cf. A137288 (positions of terms > 1).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Escape clause deleted from definition by Jianing Song, Nov 24 2021
|
|
STATUS
|
approved
|
|
|
|