3. MORE PROPERTIE S O F ADMISSIBLE ARRAYS 15

For j 1, define sets Sj and integers Tj by

Tj

=r(S\ JU1 Si), where SQ = 0, (3.19)

and

Sj = {peS\ 'u 1 Si | gcd(p, q)\rj for all q G 5, q ^ p}. (3.20)

By construction the sets Sj are pairwise disjoint, Tj r^ for j k and there exists

an integer \i so that S = U^=1 Sj. One can verify

Sj =

{p€S\JUQSi\r({p})=rj}

and use this fact to verify that rj\s for all s € Sj.

For example, if S = {24,32,40,44,46}, then n = 2 and Si = {46}, r2 = 4 and

S2 = {44}, r3=S and S3 = {24,32,40}.

To apply the generalized condition C we set, if r\ = 1, Q = Si and R = U^=2Sj

and if r± 1, Q = 0 and R = U^=15j. It is easy to see that condition (1) and

(3) of Definition 3.4 are satisfied and to test the generalized condition C, with this

collection of subsets, it remains to verify conditions (2) and (4) of Definition 3.5. In

the computer program (see Appendix A) that verifies the generalized condition C

for a set S we shall make use of this standard construction for the sets Sj. Another

application is the following theorem.

THEOREM

3.7. Let S C E be array-admissible for n and assume {in the nota-

tion of Remark 3.2) that BpC\Bq ^ 0 for all p,q e S. Write S = U^=1Sj according

to the standard decomposition, where Sj is given by (3.20) and Tj by (3.19). Set

ro = 1. //

rj-i\rj forl JV,

then lcm(S) e P(n).

PROOF .

First note that from the properties of admissible arrays it follows that

gcd(S) 1. Therefore Tj 1 for 1 j /i and we can set Q = 0 and R = \jf=1Sj.

Define integers lj by Tj = IjTj-i, 1 j \±. Since BPD Bq j^ $ for all p, q G S it

follows from Corollary 3.4 that

Y^ — 1 where

Sj

= \Sj\. (3.21)

i = i rj

We claim that there exists a sequence of nonnegative integers ij, 1 j /J with

7o = 1 such that

Sj = lj-ilj - 7j, for 1 3 V- (3-22)

If j = 1, inequality (3.21) implies s\ r\ l\ and so we can choose 71 0 such

that si = 70Z1 —71- By induction, assume that there exist nonnegative integers jj