

A087788


3Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = p*q*r, where p < q < r are primes such that a^(k1) == 1 (mod k) if a is prime to k.


69



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 terms of this sequence up to x. HeathBrown proves that, for any e > 0, there are O(x^(7/20 + e)) terms of this sequence up to x.  Charles R Greathouse IV, Nov 19 2012
All 3term Carmichael numbers can be represented by certain Chernick polynomials, whose values obey a strict sdecomposition (A324460) besides certain exceptions. Under Dickson's conjecture, "almost all" 3term Carmichael numbers are primary Carmichael numbers (A324316) in the sense that C'_3(x)/C_3(x) > 1 as x > infinity, where C_3(x) counts the 3term Carmichael numbers and C'_3(x) counts the 3term primary Carmichael numbers up to x. A primary Carmichael number m has the unique property (*) that for each prime divisor p of m, the sum of the basep digits of m equals p. As a nontrivial result, all 3term Carmichael numbers m have at least the property that (*) holds for the greatest prime divisor p of m, see Kellner 2019.  Bernd C. Kellner, Aug 03 2022


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.
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269274.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1,
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.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of basep digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Math Reference Project, Carmichael Numbers
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Rosetta Code, Programs for finding 3Carmichael numbers


FORMULA

k is composite and squarefree and for p prime, pk => p1k1. A composite odd number k is a Carmichael number if and only if k is squarefree and p1 divides k1 for every prime p dividing k (Korselt, 1899) k = p*q*r, p1k1, q1k1, r1k1.


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: A104016 A002997 A355039 * 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



