A162290 Let A087788(n) = p*q*r, where p

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

Offset: 1

Author

A.K. Devaraj, Jul 01 2009

Keywords

• nonn

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. - 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

Links

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

Crossrefs

Cf. A002997, A087788, A162990.

Programs

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

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