OFFSET
1,1
COMMENTS
Conjecture (1): Any Carmichael number C divisible by 7 and 17 of the form 10k+1 can be written as C = 7*17*(120n+119). We got for the tested numbers the following values for n: 478, 1174, 34348, 38350, 82660, 107110, 158658, 162028, 176746, 179278, 262918, 313570, 377788, 563974, 710950.
Conjecture (2): Any Carmichael number C divisible by 7 and 17 of the form 10k+5 can be written as C = 7*17*(120n+95). We got for the tested numbers the following values for n: 57, 3291, 28965, 357567, 451893.
Conjecture (3): Any Carmichael number C divisible by 7 and 17 of the form 10k+3 can be written as C = 7*17*(120n+47). We got for the tested numbers the following values for n: 111835, 181157, 222733.
Conjecture (4): Any Carmichael number C divisible by 7 and 17 of the form 10k+9 can be written as C = 7*17*(120n+71). We got for the tested number the following value for n: 435446.
Note: the property of being factorizable as C = p*q*(n*(p*q+1) + p*q), where p and q are prime, is not derived from Korselt's criterion, and is not shared by all Carmichael numbers; e.g., for the Carmichael number 340561 = 13*17*23*67, even though it is of the form 10k+1, we have (23*67) mod (13*17+1) = 209 != 221 = 13*17.
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=825265, lim, 5712, if(Korselt(n), listput(v, n))); Vec(v)
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, May 04 2012
EXTENSIONS
a(15) corrected by Charles R Greathouse IV, Oct 02 2012
STATUS
approved