A394594 The largest positive integer for which there exist n arithmetic progressions whose union covers all smaller positive integers, but miss a(n) and a(n)+1.

2, 3, 5, 11, 17, 23, 36, 54, 89, 119, 179, 239, 360, 495, 719

Offset

1,1

Comments

By an explicit construction, a(2k-1)>=2^(k+1)-1, a(2k)>=3*2^k-1. From the result of Crittenden and Vanden Eynden it follows that for k>=2, a(k)<2^k. Numerical values support conjectural bounds c*2^(k/2)<a(k)<C*2^(k/2).

References

P. Balister, B. Bollobas, R. Morris, J. Sahasrabudhe, and M. Tiba, Covering intervals with arithmetic progressions, Acta Math. Hungar. 161(1) (2020), 197--200.

R.B. Crittenden and C.L. Vanden Eynden, Any n arithmetic progressions covering the first 2^n integers cover all integers, Proc. Amer. Math. Soc. 24 (1970), 475--481.

Links

Table of n, a(n) for n=1..15.

Giedrius Alkauskas, Covering systems of [1,M-1] which avoid M and M+1.

Example

8 arithmetic progressions (1,4), (1,5), (2,3), (3,8), (3,9), (4,6), (6,9), (7,11) cover all integers in the interval [1,53], miss 54 and 55. Moreover, 54 is the largest such number.

Crossrefs

Cf. A000079, A275489, A390012.

Sequence in context: A049553 A049595 A258039 * A107438 A211204 A023206

Adjacent sequences: A394591 A394592 A394593 * A394595 A394596 A394597

Keyword

nonn,more

Author

Giedrius Alkauskas, Mar 26 2026

Status

approved