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%I #31 Mar 06 2023 10:33:33
%S 1729,2821,6601,8911,15841,41041,52633,63973,101101,126217,172081,
%T 188461,670033,748657,825265,838201,997633,1033669,1082809,1773289,
%U 2628073,4463641,4909177,6840001,7995169,8719921,8830801,9585541,9890881
%N Carmichael numbers divisible by 7.
%C 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).
%C 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?
%C The conjecture follows from Korselt's criterion. - _Charles R Greathouse IV_, Oct 02 2012
%H Charles R Greathouse IV, <a href="/A182208/b182208.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%t CarmichaelNbrQ[n_] := ! PrimeQ[n] && Mod[n, CarmichaelLambda@ n] == 1; 7 Select[ Range[2500000], CarmichaelNbrQ[ 7#] &] (* _Robert G. Wilson v_, Aug 24 2012 *)
%o (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
%o forstep(n=49,1e6,42,if(Korselt(n),print1(n", "))) \\ _Charles R Greathouse IV_, Oct 02 2012
%Y Intersection of A002997 (Carmichael) and A008589 (multiples of 7). - _Michel Marcus_, Oct 11 2016
%K nonn
%O 1,1
%A _Marius Coman_, Apr 18 2012
%E Corrected by _Robert G. Wilson v_, Aug 24 2012
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