%I #18 Oct 27 2018 10:15:10

%S 2,4,6,8,16,18,30,32,40,54,64,126,128,150,162,200,224,256,486,512,750,

%T 882,1000,1024,1458,1568,1782,1950,2048,2600,2912,3750,4096,4374,5000,

%U 5632

%N Almost 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. If modulo N one number is uncovered then we speak about an almost minimal covering number.

%C We denote by T(N) the number of divisors of N. We denote by R(N) the number of uncovered numbers modulo N. Suppose N=p^k.M, where gcd(p,M)=1, p prime, R(M) = 1 and T(M) = p-1 then R(N) = 1 as well. R(p) = p-1.

%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 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>.

%e 30 is an almost minimal covering number since 1 mod 2; 2 mod 3; 4 mod 5; 4 mod 6; 8 mod 10; 12 mod 15 and 6 mod 30 covers all numbers modulo 30 except 30-folds.

%Y Cf. A160559

%K nonn,more

%O 1,1

%A _Matthijs Coster_, May 19 2009

%E Corrected by _Eric Rowland_, Oct 24 2018