OFFSET
1,1
COMMENTS
Let n be a natural number coprime to 10 and let c be the "cycle length of n" (defined below).
Conjecture 1: If n-1=2^x*c for some x<5, then n is prime. If x > 4, the relative density of primes in such numbers is 1.
Conjecture 2: If the period of the decimal expansion of 1/n is n-1 or a divisor of n-1, and if n-1=2^x*c or n+1=2^x*c for some x, then n is prime.
- Lear Young, with contributions from Peter Košinár, Giovanni Resta, Charles R Greathouse IV, May 22 2013
To define the "cycle length of n" (using n=73 as an example):
Step 1 : 73 + 1 = 74. Get the odd part of 74, which is 37
Step 2 : 73 + 37 = 110. Get the odd part of 110, which is 55
Step 3 : 73 + 55 = 128. Get the odd part of 128, which is 1
Continuing this operation (with 73+1) repeats the same steps as above. There are 3 steps in the cycle, so the cycle length of 73 is c=3. (same to A179382((73+1)/2)=3).
More for the "cycle length of n" see link and cross references.
The numbers in the sequence are the values of n in the above conjecture when c=4 in case (1).
LINKS
Lear Young and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 117 terms from Young)
Hagen von Eitzen, Details of the "cycle length of n"
EXAMPLE
(1217-1)/16 = 76 = A179382(609).
PROG
(PARI) oddres(n)=n>>valuation(n, 2)
cyc(d)=my(k=1, t=1); while((t=oddres(t+d))>1, k++); k
forstep(n=17, 1e4, [32, 16], if(cyc(n)==n>>4 && isprime(n), print1(n", ")))
\\ Charles R Greathouse IV, May 15 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Lear Young, May 15 2013
EXTENSIONS
Edited by Charles R Greathouse IV, Nov 11 2014
STATUS
approved