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A087788
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3-Carmichael numbers: Carmichael numbers equal to the product of 3 primes: k = p*q*r, where p < q < r are primes such that a^(k-1) == 1 (mod k) if a is prime to k.
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69
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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
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1,1
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COMMENTS
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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. Heath-Brown 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 3-term Carmichael numbers can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions. Under Dickson's conjecture, "almost all" 3-term 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 3-term Carmichael numbers and C'_3(x) counts the 3-term 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 base-p digits of m equals p. As a nontrivial result, all 3-term 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
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REFERENCES
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O. Ore, Number Theory and Its History, McGraw-Hill, 1948, Reprinted by Dover Publications, 1988, Chapter 14.
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LINKS
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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. 151-161.
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269-274.
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. 883-90.
D. R. Heath-Brown, Carmichael numbers with three prime factors, Hardy-Ramanujan Journal 30 (2007), pp. 6-12.
G. Jaeschke, The Carmichael numbers to 10^12, Math. Comp., 55 (1990), 383-389.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p 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 3-Carmichael numbers
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FORMULA
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k is composite and squarefree and for p prime, p|k => p-1|k-1. A composite odd number k is a Carmichael number if and only if k is squarefree and p-1 divides k-1 for every prime p dividing k (Korselt, 1899) k = p*q*r, p-1|k-1, q-1|k-1, r-1|k-1.
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EXAMPLE
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a(6)=6601=7*23*41: 7-1|6601-1, 23-1|6601-1, 41-1|6601-1, i.e., 6|6600, 22|6600, 40|6600.
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PROG
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(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*r-1)%(p-1)||(p*r-1)%(q-1)||(p*q-1)%(r-1), , listput(v, p*q*r))))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Nov 19 2012
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CROSSREFS
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Cf. A002997, A162290.
Sequence in context: A104016 A002997 A355039 * A173703 A306338 A300629
Adjacent sequences: A087785 A087786 A087787 * A087789 A087790 A087791
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KEYWORD
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easy,nonn
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AUTHOR
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Miklos Kristof, Oct 07 2003
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EXTENSIONS
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Minor edit to definition by N. J. A. Sloane, Sep 14 2009
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STATUS
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approved
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