OFFSET
1,1
COMMENTS
We get Carmichael numbers for n = 3, 93, 4909, 7051, 13083, 18003, 25435, 27591, 30363, 32295, 53043, 53563, 60475, 72263, 103254, 112611, 126051, 134843, 137343, 155535, 228075, 230001, 258027.
Conjecture: Any Carmichael number C divisible by 23 and 67 can be written as C = 23*67*(66n+23).
Checked for the first 23 Carmichael numbers divisible by 23 and 67.
Note: the possibility to can be written as C = a*b*(n*(b-1) +a), where a and b prime divisors of C, is a property of only some of Carmichael numbers, thus is not derived from Korselt's criterion (for instance, for the Carmichael number 29341 = 13*37*61, we have 61 mod 36 = 25). In fact, seems to be rather a property of some pairs of primes (other pair of primes which generates Carmichael numbers of this form is 41 and 241).
The conjecture follows from Korselt's criterion: 67 | a(n) so a(n) = 1 (mod 66). - Charles R Greathouse IV, Oct 02 2012
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. W. Weisstein, Carmichael Number
PROG
(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
list(lim)=my(v=List()); forstep(n=340561, lim, 101706, if(Korselt(n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jul 05 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, May 03 2012
STATUS
approved