# A062775 / this third version of formula is most coherent / put it in OEIS?

Gottfried Helms helms at uni-kassel.de
Sat Dec 20 18:29:05 CET 2003

```Gottfried Helms schrieb:
(...)
-----------------------------

In my previous posts I described the generation rule of

A =  A062775

for any A(i) as

A(i) = i * g(p,a) * g(q,b) * g(r,c) * ...

with p^a,q^b,r^c the powers of the distinct primefactors of i,
where g(p,a) was written as a formula with 3 "if-then"-constructs
to obtain a result depending on special values of the parameters.

Here is a modified version, which reflects the case-distinction in the
formula itself, and thus is a single formula for all inputs:

with

i = p^a * q^b * r^c .... where (p,q,r...) are distinct primes and

a,b,c are their powers>0 , A can be generated by the following formula:

[1]    A(i) = h(p,a) * h(q,b) * h(r,c) *...

where the function h is either explained by definition [2a] or by the
definitions [2b].

Definition 1 of function h: (p is required to be prime) -----------------------------

[             3k     3 + (-1)^k  ]
[             --  -  ----------  ]   [          p  ]
2k      [  2k-1       2          4       ]   [  3 + (-1)   ]
[2a]    h(p,k) = p     +  [ p       - p                    ] * [ ---------   ]
[                                ]   [     2       ]

... or, with the substitution for the 3 + (-1)... term

Definition 2 of function h: (p is required to be prime)   --------------------------

3 + (-1)^j
[2b.1]    t(j) = ------------
2

2k     [ 2k-1      0.5*(3k - t(k) ) ]
[2b.2]    h(p,k) = p     + [p       - p                 ] * t(p)

----------------------------------

I think, formula [1], [2b.1] and [2b.2] are now concise enough to put them
into the sequence description as maple/mathematica-code for the OEIS-entry.

Could someone kindly provide a translation for that purpose?

-----------------------------------

From experimenting with exponents higher 2 in the basic problem ("number
of pythagorean triples...") it seems, that the generating-formula for
the exponent 2 is only by chance so tightly related to the primes; for
exponent 3 the primality is -at least- no obvious aspect in the list of
the results, so possibly the above formula will be revised eventually,
if a more general solution can be found and a more general algorithm has
to be reflected with the formula.

Gottfried Helms

```