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%I #14 Nov 29 2016 01:22:33
%S 7,23,48,22,47,45,45,21,44,163,162,43,161,280,1684,1363,159,351,950,
%T 1675,1358,949,158,345,1829,947,1353,510,938,1660,2796,1820,820,10208,
%U 2779,935,1650,817,937,1822
%N Let A087788(n) = p*q*r, where p<q<r, be the n-th 3-Carmichael number. Then a(n) = (p-1)*(p*q*r-1)/((q-1)*(r-1)).
%C A.K. Devaraj conjectured that a(n) is always an integer, and this was proved by Carl Pomerance.
%C a(n) may be called the Pomerance index of the n-th 3-Carmichael number.
%C An application of Pomerance index: The index for the Carmichael number 561 is 7. This can be used to prove that 561 is the only 3-factor Carmichael number with 3 as one of its factors. Proof: Let N be a 3-factor composite number. Keep 3 fixed and increase the other two prime factors indefinitely. The relevant Pomerance index is a number less than 7 but greater than 6. As the other two prime factors are increased indefinitely the Pomerance index becomes asymptotic to 6. Hence 561 is the only 3-factor Carmichael number with 3 as a factor. - _A.K. Devaraj_, Jul 27 2010
%C Let p be a prime number. Then, along the lines indicated above, it can be proved that there are only a finite number of 3-Carmichael numbers divisible by p. - _A.K. Devaraj_, Aug 06 2010
%H Charles R Greathouse IV, <a href="/A162290/b162290.txt">Table of n, a(n) for n = 1..10000</a>
%o (PARI) do(lim)=my(v=List()); forprime(p=3, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), forprime(r=q+1, lim\(p*q), if((q*r-1)%(p-1)||(p*r-1)%(q-1)||(p*q-1)%(r-1), , listput(v, [p*q*r,(p*q*r-1)*(p-1)/(q-1)/(r-1)]))))); v=vecsort(v,1); vector(#v,i,v[i][2]) \\ _Charles R Greathouse IV_, Sep 07 2016
%Y Cf. A002997, A087788, A162990.
%K nonn
%O 1,1
%A _A.K. Devaraj_, Jul 01 2009
%E Edited by _N. J. A. Sloane_, Sep 14 2009, based on email messages from _David Broadhurst_ and _M. F. Hasler_, Jul 10 2009
%E Spelling corrected by _Jason G. Wurtzel_, Aug 23 2010
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