OFFSET
1,1
COMMENTS
The following Carmichael numbers are of the form n*(2n-1)*(3n-2)*(6n-5): 63973, 31146661, 703995733, 2414829781, 192739365541, 461574735553, 3976486324993.
The following Carmichael numbers are of the form n*(2n-1)*(5n-4)*(10n-9): 172081, 881936608681, 3307287048121, 8916642713161.
The following Carmichael number is of the form n*(2n-1)*(7n-6)*(14n-13): 167979421.
The following Carmichael number is of the form n*(2n-1)*(9n-8)*(18n-17): 277241401.
The following Carmichael number is of the form n*(2n-1)*(11n-10)*(22n-21): 9924090391909.
The following Carmichael number is of the form n*(2n-1)*(15n-14)*(30n-29): 7932245192461.
The following Carmichael number is of the form n*(2n-1)*(17n-16)*(34n-33): 3145699746793.
The following Carmichael numbers are of the form n*(2n-1)*(21n-20)*(42n-41): 1504651681, 117765525241, 2732745608209.
The following Carmichael number is of the form n*(2n-1)*(23n-22)*(46n-45): 7066238244481.
For p=13 and p=19, there is no Carmichael number up to 10^13.
There is not any other Carmichael number of this form, for p from 3 to 23, up to 10^13.
Conjecture: for any odd number p we have an infinite number of Carmichael numbers of the form n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1).
Note: many numbers of the form n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1), not divisible by 2, 3 or 5, where p is odd or even, are squarefree and respects the Korselt's criterion for many of their prime divisors or are not squarefree but respects the Korselt's criterion sometimes even for all their divisors (but we didn’t find Carmichael numbers when p is even).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. W. Weisstein, Carmichael Number
E. W. Weisstein, Korselt’s Criterion
PROG
(PARI) Kv(n, v)=for(i=2, #v, for(j=1, i-1, if(gcd(v[i], v[j])>1, return(0)))); for(i=1, #v, my(f=factor(v[i])); for(j=1, #f~, if(f[j, 2]>1 || (n-1)%(f[j, 1]-1), return(0)))); 1
list(lim)=my(v=List(), n, C); forstep(p=3, 23, 2, n=3; while((C=n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1))<=lim, if(Kv(C, [n, 2*n-1, p*n-p+1, 2*p*n-2*p+1]), listput(v, C)); n+=2)); Set(v) \\ Charles R Greathouse IV, Jul 07 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, May 29 2012
EXTENSIONS
a(8) and a(10) inserted by Charles R Greathouse IV, Jul 07 2017
STATUS
approved