

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


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

Cf. A002997, A087788, A162990.
Sequence in context: A158035 A101789 A174590 * A180044 A062725 A147121
Adjacent sequences: A162287 A162288 A162289 * A162291 A162292 A162293


KEYWORD

nonn


AUTHOR

A.K. Devaraj, Jul 01 2009


EXTENSIONS

Edited by N. J. A. Sloane, Sep 14 2009, based on email messages from David Broadhurst and M. F. Hasler, Jul 10 2009
Spelling corrected by Jason G. Wurtzel, Aug 23 2010


STATUS

approved



