n such that there is one sequence of length n having equal sum and product
Don Reble
djr at nk.ca
Fri Apr 8 01:41:38 CEST 2005
> I think it is also worth mentioning the minimal constraint that terms > 2
> in A033179 must be of the form p+1 (since [ a+1, b+1, <1> x (ab - 1) ]
> is always a solution for k = ab+1, and non-trivial when a >= b >= 2).
Also, 2L-1 must not be composite; if it has proper factorization a*b, then
1*(L-3), 2, (a+1)/2, (b+1)/2
is a second solution for length L.
Combining those ideas with those from Louis Marmet, we see that:
For L-1 and 2L-1 to be non-composite, L == 0,12,24 mod 30; or L=2,3,4,6.
But if L = 30n+12, then
1*(30n+7), 2, 2, 2, 2, 2n+1
is a second solution.
So all entries in the sequence are 2,3,4,6, or 0,24 mod 30.
One can eliminate some of the multiples of 5 with such congruences;
for example,
1*(35n+25),2,2,3,3,n+1 eliminates L=35n+30,
1*(35n+22),4,9,n+1 eliminates L=35n+25
1*(55n+37),4,14,n+1 eliminates L=55n+40
But it may be hard to find a complete set of congruences.
--
Don Reble djr at nk.ca
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