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
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.
Michael Penn, Romanian Mathematical Olympiad Problem, Youtube video, 2020.
Wikipedia, Cunningham chain
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:
end: # Lorenzo Sauras Altuzarra, Feb 12 2021
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
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 17 2010
EXTENSIONS
Escape clause added to definition by N. J. A. Sloane, Feb 19 2021
Escape clause deleted from definition by Jianing Song, Nov 24 2021
STATUS
approved