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%I #26 Mar 06 2023 10:57:01
%S 2821,15841,75361,172081,399001,512461,852841,1193221,1857241,2100901,
%T 2113921,3146221,4903921,5049001,5481451,6049681,8341201,8927101,
%U 9585541,10606681,10837321,11205601,18162001,27062101,27402481
%N Carmichael numbers divisible by 31.
%C Conjecture: Any Carmichael number C divisible by 31 can be written in one of the following three forms: C = 2790n+31; C = 2790n+31^2 or C = 2790n+31*61, where n is natural.
%C Examples:
%C Carmichael numbers of the first form: 2821, 75361, 399001, 2100901.
%C Carmichael numbers of the second form: 6049681, 10837321, 11205601.
%C Carmichael numbers of the third form: 15841, 172081, 512461, 852841, 1193221, 1857241, 2113921, 3146221, 4903921, 5049001, 5481451, 8341201, 8927101, 9585541, 10606681, 18162001.
%C Checked for the first 23 Carmichael numbers divisible by 31.
%C It follows from Korselt's criterion that such numbers are 31 mod 930, the union of the three residue classes. Thus the conjecture is true. - _Charles R Greathouse IV_, Oct 02 2012
%H Charles R Greathouse IV, <a href="/A182151/b182151.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; 31 Select[ Range@ 1000000, CarmichaelNbrQ[ 31#] &] (* _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=961,1e5,930,if(Korselt(n),print1(n", "))) \\ _Charles R Greathouse IV_, Oct 02 2012
%K nonn
%O 1,1
%A _Marius Coman_, Apr 18 2012
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