For k \geq 0, the authors prove that p(k)(n) is positive for all sufficiently large positive integers n if and only if A has the property Pk, viz. there are more than k elements in A and, if we remove an arbitrary subset of k elements from A, the remaining elements have greatest common divisor unity. Among a number of other results we may select as typical: For any k, let A infinite and have property Pk and let n > oo. Then p(k) (n) n-c > +oo for any fixed c. Also, if
then \rho(k)(n) > 0, n\rho(k)(n) is unbounded above and n\rho(k-1)(n) > +oo.
Classif.: * 11P82 Analytic theory of partitions
Index Words: Number Theory
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