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A162290
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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)).
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5
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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
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1,1
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COMMENTS
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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 n-th 3-Carmichael 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 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
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
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LINKS
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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