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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 276.05005

**Autor: ** Abbott, H.L.; Erdös, Paul; Hanson, D.

**Title: ** On the number of times an integer occurs as a binomial coefficient. (In English)

**Source: ** Am. Math. Mon. 81, 256-261 (1974).

**Review: ** Let N(t) denote the number of times the integer t < 1 occurs as a binomial coefficient; that is, N(t) is the number of solutions of t = \binom{n}{r} in integers n and r. In this note we obtain some additional information about the behavior of N(t). In Theorem 1 we prove that the average and normal order of N(t) is 2; in fact, we prove somewhat more than this, namely, the number of integers t, 1 < t \leq x, for which N(t) > 2 is 0(\sqrt x). [see *G. H. Hardy* and *E. M. Wright*, ``Introduction to the theory of numbers'' (1960; Zbl 086.25803), p. 263 and p. 356, for the definitions of averageand normal order.] In Theorem 2 we give an upper bound for N(t) in terms of the number N(t) of distinct prime factors of t: N(t) < 2w(t) log t/(log t-w(t) log log t). Our main result is Theorem 3, in which we show that N(t) = 0(log t/ log log t). Finally, in Theorem 4, we consider the related problem of determining the number of representations of an integer as a product of consecutive integers.

**Classif.: ** * 05A10 Combinatorial functions

11A41 Elemementary prime number theory

05A15 Combinatorial enumeration problems

11B39 Special numbers, etc.

11M99 Analytic theory of zeta and L-functions

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