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Statement

Let $N>0$. If congruence classes with distinct moduli $d_i>1$, each dividing $N$, cover a full period of length $N$, then

$$2N\le\sigma_1(N),\qquad \sigma_1(N)=\sum_{d\mid N}d.$$

In particular the least common multiple $L$ of the moduli of any finite covering of $\mathbb Z$ with distinct moduli greater than one is non-deficient. If all the moduli are odd, $L$ is odd. Here non-deficient means perfect or abundant; the conclusion is not automatically strict abundance.

Complete proof

By Lemma 4.1, $N\le\sum_i N/d_i$. Distinctness makes the $d_i$ a subset of the divisors of $N$ other than $1$. Hence

$$N\le\sum_i\frac N{d_i} \le\sum_{d\mid N,\ d>1}\frac Nd =\sigma_1(N)-N.$$

The last identity follows because $d\mapsto N/d$ is an involution of the positive divisors, and the removed divisor $1$ contributes $N$. Rearranging proves the claim.

For an integer covering, $L>0$ because it is the least common multiple of positive integers; every modulus divides it. The periodicity lemma reduces to its finite period. A covering has a nonempty index set, although the empty least common multiple convention $L=1$ is harmless. If every modulus is odd, their product is odd and is divisible by $L$. Thus $L$ is odd too.

Source and dependencies

Canonical v1, p. 5, Lemma 4.2 and §4.2. Complete deduction from Lemma 4.1 and elementary divisor arithmetic. The authors describe this density observation as folklore, without a novelty claim.

Bears on

• Problem 7.