

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



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



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



