A160559 Minimal covering numbers.

12, 80, 90, 210, 280, 378, 448, 1386, 1650, 2200, 2464, 5346, 9750, 11264, 11466, 13000, 14994, 18954, 20384, 23166, 26656, 27846, 30294, 31122, 33150, 33858, 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

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.

Jai Setty, Covering systems corresponding to the terms a(1)-a(87).

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

Cf. A160560.

Sequence in context: A243955 A232044 A190216 * A038734 A258591 A058962

Adjacent sequences: A160556 A160557 A160558 * A160560 A160561 A160562

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