Problems with A076107-9

[David Wilson]

The following observations were brought to my attention by T. D. Noe.

A076107 purports to be "The smallest of n consecutive numbers whose sum is 
an nth power, or 0 if no such number exists".

We cannot take "number" to mean "integer", for then there are arbitrarily 
large negative nth powers for odd n which are the sum of n consecutive 
integers, so the described value does not exist for odd n and we should have 
a(n) = 0 for odd n.

We could take n to mean "nonnegative integer".  Then if a(n) = 0 would be 
ill-defined in that it would not be apparent whether 0 were the first term 
of the sum or the sum did not exist.  Also, a(1) = 0, which disagrees with 
the published a(1) = 1.

We could also take n to mean "positive integer".  Then a(n) = 0 always means 
that the sum does not exist.  But a(2) = 4, which disagrees with the 
published value a(2) = 0.

I lean towards the "positive integer" interpretation, because a(n) = 0 is 
then unambiguous.  Either interpretation requires that A076107-9 be 
corrected.

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A076108 purports to be the "Smallest n-th power which is a sum of n 
consecutive natural numbers or 0 if no such number exists."  Instead of 
"number" in A076107, the equally ambiguous term "natural number" is used. 
It is reasonable to presume that the sums mentioned in A076107 and A076108 
are the same for corresponding terms, so that the two sequences would be 
related by

    A076108(n) = n*A076107(n) + n*(n-1)/2.

Also note that the published value A076108(2) = 1 disagrees with the comment 
on A076108 that "a(2) = 9 = 4+5".  The comment is consistent with the 
"positive integer" interpretation.

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Finally, A076109 is defined as "a(n) = A076108(n)^(1/n)."  As it stands, it 
appears to be multiplicative.  However, the required modifications to 
A076107-9 will make it nonmultiplicative, which is a shame.


- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
   - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"