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is any sequence of rational integers satisfying (1.1), then after a certain point the sequence becomes periodic when considered modulo m. Its least period is called the characteristic number of the sequence (U) modulo m. In a recent paper in these Transactionst, I have considered the problem of determining this characteristic number given m, C, C2, , * * Ck and the k initial values UO, U1, *, Uk-i of the sequence (U), and I have reduced it to certain basic problems in the theory of higher congruences.$ In the present paper, I am concerned with the following problem which I shall similarly reduce to a problem in the theory of higher congruences: Given any positive integer s: to find the number of distinct sequences (U) modulo m whose characteristic number is exactly equal to s. 2. I obtain here the following results.

An enumerative problem in the arithmetic of linear recurring series