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A182208
Carmichael numbers divisible by 7.
1
1729, 2821, 6601, 8911, 15841, 41041, 52633, 63973, 101101, 126217, 172081, 188461, 670033, 748657, 825265, 838201, 997633, 1033669, 1082809, 1773289, 2628073, 4463641, 4909177, 6840001, 7995169, 8719921, 8830801, 9585541, 9890881
OFFSET
1,1
COMMENTS
Conjecture: Any Carmichael number C divisible by 7 can be written in one of two ways: (1) C=7*(6m+1)*(6n+1), where m and n are natural numbers or (2) C=7*(6m-1)*(6n-1), where m and n are natural numbers. In other words, there aren’t Carmichael numbers divisible by 7 of the form C=7*(6m+1)*(6n-1). Checked for the first 27 Carmichael numbers divisible by 7. Note: a Carmichael number with more than 3 prime divisors can be written (sometimes) in both ways: 41041 = 7*11*13*41 = 7*13*451 (form 1) = 7*11*533 = 7*41*143 (form 2).
Observation: in the first 100 Carmichael numbers with three prime divisors (not divisible by 3) there is no one to can be written as (6x+1)(6y+1)(6z-1), they are all of the form (6x+1)(6y+1)(6z+1), (6x-1)(6y-1)(6z-1) or (6x+1)(6y-1)(6z-1). Would not that be enough to make an assumption that there are no such Carmichael numbers with three prime divisors, or even more, that aren't Carmichael numbers even with more than three divisors to can be written this way?
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
Eric Weisstein's World of Mathematics, Carmichael Number
MATHEMATICA
CarmichaelNbrQ[n_] := ! PrimeQ[n] && Mod[n, CarmichaelLambda@ n] == 1; 7 Select[ Range[2500000], CarmichaelNbrQ[ 7#] &] (* Robert G. Wilson v, Aug 24 2012 *)
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
forstep(n=49, 1e6, 42, if(Korselt(n), print1(n", "))) \\ Charles R Greathouse IV, Oct 02 2012
CROSSREFS
Intersection of A002997 (Carmichael) and A008589 (multiples of 7). - Michel Marcus, Oct 11 2016
Sequence in context: A198775 A154729 A083737 * A340092 A324316 A182207
KEYWORD
nonn
AUTHOR
Marius Coman, Apr 18 2012
EXTENSIONS
Corrected by Robert G. Wilson v, Aug 24 2012
STATUS
approved