Are A055999 and A074171 somehow the same?

[Dean Hickerson]

Alonso Del Arte wrote:

> A055999 is a(n)=n*(n+7)/2, which works out to be ((n^2 - n)/2) - 3n
> with a different offset. A074171 is Start with 1, add the next number
> if one gets a prime then subtract the next number else add the next. I
> entered the two sequences into Mathematica, omitting the first two
> terms of A074171, then asked A055999 == A074171 and it answered True.
>
> So the question I have now is: has anyone else noticed this and if so
> have they been able to determine whether this is just a huge
> coincidence because they've found that the two sequences diverge at a
> larger n or do the two sequences agree for as far as has been checked?

Jud McCranie replied:

> I'm unable to understand how A74171 is generated.  For instance, with
> a(4)=4, add the next number, 5, gives 9, composite, so 4+5 = 9 is the next
> term.  Then a(5)=9, add the  next number (10?), gives 19, prime, so
> subtract 10 from 9?  What am I doing wrong?

For A074171, a(n) is either a(n-1)+n or a(n-1)-n; we only use the minus
sign if a(n-1) is prime.  E.g. since a(2)=3 is prime, a(3)=a(2)-3=0.

In fact  a(n) = (n-3)(n+4)/2  for  n >= 3.  The proof is by induction:
For  n>3,  a(n-1) = (n-4)(n+3)/2  is composite, so  a(n) = a(n-1) + n =
(n-3)(n+4)/2.

So the two sequences are the same, except for the first two terms.

I recommend that A074171 be deleted, unless someone else wants to spend the
time to clarify the description.

Dean Hickerson
dean at math.ucdavis.edu