A087788 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.

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

Offset: 1

Author

Miklos Kristof, Oct 07 2003

Keywords

• easy
• nonn

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. 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

Examples

			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.
		

References

• O. Ore, Number Theory and Its History, McGraw-Hill, 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. 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

Crossrefs

Intersection of A002997 and A007304.

Cf. A162290.

Programs

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

Formula

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.

Extensions

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