OFFSET
1,1
COMMENTS
Note that in this sequence, 60k+13, 180k+37, and 300k+61 do not have to be prime.
Conjecture: N = (60k+13)*(180k+37)*(300k+61) is a Carmichael number if (but not only if) 60k+13, 180k+37 and 300k+61 are all three prime numbers.
We checked the conjecture up to k = 175; we got Carmichael numbers with three prime divisors for k = 0, 6, 7, 11, 14, 28, 60, 62, 80, 102, 126, 137, 139, 157, 171.
We got Carmichael numbers with more than three prime divisors for k = 2, 59, 67, 82.
Note: we can see that 13 = 7*2-1, 37 = 7*6-5 and 61 = 7*10-9, while 60 = 30*2, 180 = 30*6 and 300 = 30*10; we also have Carmichael numbers that can be written as (30n-11)*(60n-23)*(150k-59), for instance 63973, or as (30n-7)*(90n-23)*(300k-79), for instance 340561; we can see that, this time, 23 = 11*2+1, 59 = 11*5+4, 23 = 7*3+2 and 79 = 7*10+9, while 60 = 30*2, 150 = 30*5, 90 = 30*3 and 300 = 30*10.
Observation: many Carmichael numbers, not only with three prime divisors, can be written in one of the following two forms: C = ((30*a*m - (a*p+a-1))*((30*b*m - (b*p+b-1))*((30*c*m - (c*p+c-1)) or C = ((30*a*m + (a*p-a+1))*((30*b*m + (b*p-b+1))*((30*c*m + (c*p-c+1)), where p, a*p+a-1, b*p+b-1 and c*p+c-1 are all (four or three, if a=1)primes (in the first case) or p, a*p-a+1, b*p-b+1 and c*p-c+1 are all primes (in the second case).
The conjecture follows from Korselt's criterion. - 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) test(lim)={
my(v=List(), n, f);
for(k=0, lim,
n=(60*k+13)*(180*k+37)*(300*k+61)-1;
f=factor(60*k+13);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(180*k+37);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(300*k+61);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
listput(v, n+1)
);
Vec(v)
}; \\ Charles R Greathouse IV, Oct 02 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, Apr 28 2012
STATUS
approved