Some Extremal Problems on Divisibility Properties of Sequences of Integers

Note that, by (4 .4) and (4 .5), (7 .6) holds for all nonnegatíve p. Substituting from (7 .6) in (6 .1) and (6 .2) and evaluating coefficients of xm, we obtain the following two identities. r-1 m (p + q)T rP~+4) = pTrpm + ~,(q) + T (P)-L q r, m pq E a l i r-a-1, m-j s-0 j-0 r-1 m-1 pq ~-~ T(P)Tr(9a-l. m-i-1 a L-0 i-0 r-1 m (r + i)Tr m+q) _ (r + 1)Tr + p (r-s)T (P)T (q) a. j r-8-1, M-j 8-0 j-0 r-1 m-1 p E (r-S)T(P)T(q) e .j r-8-1 .m-d-1 8-0 j-0 In particular, for q = 0, (7 .8) reduces to r-1 (r + 1). . .,) _ ~ m} m 3 + 2 m(P) `~ ~(~;) 1 r-8-1 .m-j a-0 j-0 r-1 m-1 _ 2 m(P)-p S E ~~ 1r-8-1 .m-j-1 8-0 j-0 We remark that (6 .1) is implied by (6 .2). To see this, multiply both sides of (6 .2) by q, interchange p and q, and then add corresponding sides of the two equations. Similarly, it can be verified that (7 .3) is implied by (7 .4) and (7 .7) is implied by (7 .8). REFERENCE 1. E. D. Raínvílle. 3pecia? Furs tions. Dedicated to the memo .ty o5 my ,J y-tíekid Vehn Hoggatt A sequence of integers A = {a 1 < a2 <. .. < a k < n} is said to have property P,(n) if no a j divides the product of r other a's. Property P(n) means that no a j divides the product of the other a's. A sequence has property Q(n) if the products ajaj are all distinct. Many decades ago I proved the following theorems [2] : Let A have property ? 1 (i .e ., no aj divides any other). Then maxk= l n +1 J 2 The proof is easy .