

A162290


Let A087788(n) = p*q*r, where p<q<r, be the nth 3Carmichael number. Then a(n) = (p1)*(p*q*r1)/((q1)*(r1)).


5



7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 162, 43, 161, 280, 1684, 1363, 159, 351, 950, 1675, 1358, 949, 158, 345, 1829, 947, 1353, 510, 938, 1660, 2796, 1820, 820, 10208, 2779, 935, 1650, 817, 937, 1822
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OFFSET

1,1


COMMENTS

A.K. Devaraj conjectured that a(n) is always an integer, and this was proved by Carl Pomerance.
a(n) may be called the Pomerance index of the nth 3Carmichael number.
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 3factor Carmichael number with 3 as one of its factors. Proof: Let N be a 3factor 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 3factor Carmichael number with 3 as a factor.  A.K. Devaraj, Jul 27 2010
Let p be a prime number. Then, along the lines indicated above, it can be proved that there are only a finite number of 3Carmichael numbers divisible by p.  A.K. Devaraj, Aug 06 2010


LINKS



PROG

(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*r1)%(p1)(p*r1)%(q1)(p*q1)%(r1), , listput(v, [p*q*r, (p*q*r1)*(p1)/(q1)/(r1)]))))); v=vecsort(v, 1); vector(#v, i, v[i][2]) \\ Charles R Greathouse IV, Sep 07 2016


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



