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?
Thanks in advance!
-----------------------------------
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
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