A160560 Almost covering numbers.

2, 4, 6, 8, 16, 18, 30, 32, 40, 54, 64, 126, 128, 150, 162, 200, 224, 256, 486, 512, 750, 882, 1000, 1024, 1458, 1568, 1638, 1782, 1950, 2048, 2600, 2912, 3750, 4096, 4374, 5000, 5632, 6174, 6318, 8192, 10976

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. If modulo N we cannot cover all residues, but can cover all but one residue, then N is an almost covering number.

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

Some further terms that are obtained this way include: 12750, 13122, 16384, 16758, 17000, 18750, 19602, 21294, 25000, 25350, 26624, 29792, 32768, 33800, 37856, 39366, 43218, 61952, 65536, 74358, 76832, 82134, 93750. - Max Alekseyev, Feb 12 2025

Links

Table of n, a(n) for n=1..41.

Donald Jason Gibson, A covering system with least modulus 25, Math. Comp. 78, (2009), 1127-1146.

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

30 is an almost 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.

Crossrefs

Cf. A160559.

Sequence in context: A073696 A058602 A133808 * A332291 A093109 A070034

Adjacent sequences: A160557 A160558 A160559 * A160561 A160562 A160563

Keyword

nonn,more,changed

Author

Matthijs Coster, May 19 2009

Extensions

Corrected by Eric Rowland, Oct 24 2018

Edited, missing term a(27)=1638 inserted, and a(38)-a(41) added by Max Alekseyev, Feb 08 2025

Status

approved