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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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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 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. [From A. K. Devaraj (dkandadai(AT)gmail.com), Jul 27 2010]
Let p be a prime number. Then, on the lines indicated above, it can be proved that there can be only a finite number of 3-Carmichael numbers with p as one of its factors. [From A.K.Devaraj (dkandadai(AT)gmail.com), Aug 06 2010]
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CROSSREFS
| Cf. A002997, A087788, A162990.
Sequence in context: A185955 A158035 A101789 * A180044 A062725 A147121
Adjacent sequences: A162287 A162288 A162289 * A162291 A162292 A162293
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KEYWORD
| nonn
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AUTHOR
| A. K. Devaraj (dkandadai(AT)gmail.com), Jul 01 2009
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 14 2009, based on email messages from David Broadhurst and M. H. Hasler, Jul 10 2009
Spelling corrected by Jason G. Wurtzel (j_seq(AT)wurtzel.com), Aug 23 2010
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