login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A162290 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)). 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 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

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*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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 7 04:20 EDT 2020. Contains 333292 sequences. (Running on oeis4.)