Problems with A076107-9

[Max]

Interestingly these sequences have caught my eye as well.
So that I've submitted the following formulae:

%I A076108
%S A076108 1,1,27,0,3125,729,823543
%F A076108 a(4k)=0; otherwise a(n)=p1^n*...*pm^n where p1,...,pm are all distinct odd primes dividing n.
%o A076108 (PARI) {A076108(n)=if(n%4==0,return(0));m=n;if(m%2==0,m\=2);f=factorint(m)[,1];prod(i=1,length(f),f[i])^n}
%O A076108 1
%K A076108 ,nonn,

%I A076109
%S A076109 1,1,3,0,5,3,7,0,3,5
%F A076109 a(4k)=0; otherwise a(n)=p1*...*pm where p1,...,pm are all distinct odd primes dividing n.
%o A076109 (PARI) {A076109(n)=if(n%4==0,return(0));if(n%2==0,n\=2);f=factorint(n)[,1];prod(i=1,length(f),f[i])}
%O A076109 1
%K A076109 ,mult,nonn,

%I A076107
%S A076107 1,0,8,0,623,119,117646
%F A076107 a(4k)=0; otherwise a(n) = (2*A076108(n)/n - n + 1)/2 = (2*p1^n*...*pm^n/n - n + 1)/2 where p1,...,pm are all distinct odd primes dividing n.
%o A076107 (PARI) {A076107(n)=if(n%4==0,return(0));m=n;if(m%2==0,m\=2);f=factorint(m)[,1];p=1;(2*prod(i=1,length(f),f[i])^n/n-n+1)/2}
%O A076107 1
%K A076107 ,nonn,

Regards,
Max

David Wilson wrote:
> 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.
> 
> -----------------------
> 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.
> 
> ----------------------
> 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"
> 
>