%I #52 May 29 2024 20:46:44

%S 12,80,90,210,280,378,448,1386,1650,2200,2464,5346,9750,11264,11466,

%T 13000,14994,18954,20384,23166,26656,27846,30294,31122,33150,33858,

%U 36608,37050,37674,44200,44850,49400,49504,53248,53900,55328,59800,63750,66976,71250,72930,85000,95000,95744,97240,100100,107008,107406,112112,117306,120042,131274,142002,145314,192500,208544,223074,242250,252448,272272,293250,311168,318500,323000,369750,385434,391000,395250,423500,431250,450846,452608,485982,493000,505856,519498,527000,568458,575000,612352,617526,654500,660114,685216,731500,735150,747954

%N Minimal covering numbers.

%C 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.

%C Hough and Nielsen (2019) proved that each term must be divisible by 2 or 3. - _Max Alekseyev_, Nov 19 2022

%H Jai Setty, <a href="/A160559/b160559.txt">Table of n, a(n) for n = 1..87</a>

%H Max Alekseyev, <a href="/A160559/a160559_2.txt">Covering systems corresponding to the terms a(1)-a(41)</a>.

%H Donald Jason Gibson, <a href="https://doi.org/10.1090/S0025-5718-08-02154-6">A covering system with least modulus 25</a>, Math. Comp. 78 (2009), 1127-1146.

%H Robert D. Hough and Pace P. Nielsen, <a href="https://doi.org/10.1215/00127094-2019-0058">Covering systems with restricted divisibility</a>, Duke Math. J. 168:17 (2019), 3261-3295. arXiv:<a href="https://arxiv.org/abs/1703.02133">1703.02133 [math.NT]</a>.

%H Pace P. Nielsen, <a href="https://doi.org/10.1016/j.jnt.2008.09.016">A covering system whose smallest modulus is 40</a>, Journal of Number Theory 129 (2009), 640-666.

%H Pace P. Nielsen, <a href="http://www.youtube.com/watch?v=3ev1YjVl0RY">A movie explaining covering systems</a>.

%H Jai Setty, <a href="/A160559/a160559.pdf">Covering systems corresponding to the terms a(1)-a(87)</a>.

%e 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.

%e 36 is not minimal since 12 is a divisor and 12 is the LCM of a covering system.

%Y Cf. A160560.

%K nonn

%O 1,1

%A _Matthijs Coster_, May 19 2009

%E Corrected by _Eric Rowland_, Oct 24 2018

%E a(17)-a(23) from _Max Alekseyev_, Nov 19 2022

%E a(24)-a(41) from _Max Alekseyev_, Mar 21 2023

%E Missing terms a(8) and a(15) inserted and their multiples removed by _Jai Setty_, May 29 2024