

A087788


3Carmichael numbers: Carmichael numbers equal to the product of 3 primes: n=pqr, where p<q<r are primes such that a^{n1} == 1 (mod n) if a is prime to n.


67



561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 46657, 52633, 115921, 162401, 252601, 294409, 314821, 334153, 399001, 410041, 488881, 512461, 530881, 1024651, 1152271, 1193221, 1461241, 1615681, 1857241, 1909001, 2508013
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OFFSET

1,1


COMMENTS

It is interesting that most of the numbers have the last digit 1. For example 530881, 3581761, 7207201, etc.
Granville & Pomerance conjecture that there are ~ c x^{1/3}/(log x)^3 members of this sequence up to x. HeathBrown proves that, for any e > 0, there are O(x^{7/20 + e}) members of this sequence up to x.  Charles R Greathouse IV, Nov 19 2012


REFERENCES

O. Ore, Number Theory and Its History, McGrawHill, 1948, Reprinted by Dover Publications, 1988, Chapter 14.


LINKS

R. J. Mathar and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 3284 terms from Mathar)
F. Arnault, Constructing Carmichael numbers which are strong pseudoprimes to several bases, Journal of Symbolic Computation, vol. 20, no 2, Aug. 1995, pp. 151161.
Harvey Dubner, Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1).
A. Granville and C. Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), pp. 88390.
D. R. HeathBrown, Carmichael numbers with three prime factors, HardyRamanujan Journal 30 (2007), pp. 612.
G. Jaeschke, The Carmichael numbers to 10^12, Math. Comp., 55 (1990), 383389.
Math Reference Project, Carmichael Numbers
R. G. E. Pinch, Carmichael numbers up to 10^16 (FTP)
Rosetta Code, Programs for finding 3Carmichael numbers


FORMULA

n is composite and squarefree and for p prime, pn => p1n1. A composite odd number n is a Carmichael number if and only if n is squarefree and p1 divides n1 for every prime p dividing n (Korselt, 1899) n=pqr, p1n1, q1n1, r1n1.


EXAMPLE

a(6)=6601=7*23*41: 7166011, 23166011, 41166011, i.e. 66600, 226600, 406600.


PROG

(PARI) list(lim)=my(v=List()); forprime(p=3, (lim)^(1/3), forprime(q=p+1, sqrt(lim\p), forprime(r=q+1, lim\(p*q), if((q*r1)%(p1)(p*r1)%(q1)(p*q1)%(r1), , listput(v, p*q*r))))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Nov 19 2012


CROSSREFS

Cf. A002997, A162290.
Sequence in context: A309235 A104016 A002997 * A173703 A306338 A300629
Adjacent sequences: A087785 A087786 A087787 * A087789 A087790 A087791


KEYWORD

easy,nonn


AUTHOR

Miklos Kristof, Oct 07 2003


EXTENSIONS

Minor edit to definition by N. J. A. Sloane, Sep 14 2009


STATUS

approved



