OFFSET
1,1
COMMENTS
A collection of congruences with distinct moduli, each greater than 1, such that each integer satisfies at least one of the congruences, is said to be a covering system. Let N be the LCM of these moduli. We consider minimal N's, i.e., N is the LCM of some moduli, but none of the divisors has this property.
Hough and Nielsen (2019) proved that each term must be divisible by 2 or 3. - Max Alekseyev, Nov 19 2022
LINKS
Jai Setty, Table of n, a(n) for n = 1..87
Max Alekseyev, Covering systems corresponding to the terms a(1)-a(41).
Donald Jason Gibson, A covering system with least modulus 25, Math. Comp. 78 (2009), 1127-1146.
Robert D. Hough and Pace P. Nielsen, Covering systems with restricted divisibility, Duke Math. J. 168:17 (2019), 3261-3295. arXiv:1703.02133 [math.NT].
Pace P. Nielsen, A covering system whose smallest modulus is 40, Journal of Number Theory 129 (2009), 640-666.
Pace P. Nielsen, A movie explaining covering systems.
EXAMPLE
80 is in the set since 1 mod 2; 2 mod 4; 4 mod 8; 8 mod 16; 4 mod 5; 8 mod 10; 16 mod 20, 32 mod 40; 0 mod 80 is a covering system with LCM 80. None of the divisors has that property.
36 is not minimal since 12 is a divisor and 12 is the LCM of a covering system.
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthijs Coster, May 19 2009
EXTENSIONS
Corrected by Eric Rowland, Oct 24 2018
a(17)-a(23) from Max Alekseyev, Nov 19 2022
a(24)-a(41) from Max Alekseyev, Mar 21 2023
Missing terms a(8) and a(15) inserted and their multiples removed by Jai Setty, May 29 2024
STATUS
approved