Equal sum-product sequences

David W. Wilson wilson at cabletron.com
Mon Apr 20 18:21:28 CEST 1998

```For n = 1 to 10000, I computed the number f(n) of nondecreasing sequences of
n positive integers whose sum and product are equal (I have already sent the
sequence to Sloane).  I found

n    f(n)    Sequences of length n with equal sum and product

1     inf    (k) for any k >= 1
2       1    (2,2)
3       1    (1,2,3)
4       1    (1,1,2,4)
5       3    (1,1,1,2,5) (1,1,1,3,3) (1,1,2,2,2)
6       1    (1,1,1,1,2,6)
7       2    (1,1,1,1,1,2,7) (1,1,1,1,1,3,4)
8       2    (1,1,1,1,1,1,2,8) (1,1,1,1,1,2,2,3)
9       2    (1,1,1,1,1,1,1,2,9) (1,1,1,1,1,1,1,3,5)
10       2    (1,1,1,1,1,1,1,1,2,10) (1,1,1,1,1,1,1,1,4,4)

Clearly, for n >= 2, f(n) >= 1, as the seqence (1x(n-2),2,n) has length n and
equal sum and product 2n.

There seem to be only a few n for which f(n) = 1, that is, for which
(1x(n-2),2,n) is the unique sequence length n having equal sum and product
2n.  Up to 10000, these n are 2, 3, 4, 6, 24, 114, 174, and 444.  There is no
easy way to identify these n, is there?

```