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?
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