Modifications for A069623, added formula

Antti Karttunen karttu at
Mon Oct 14 16:05:32 CEST 2002

Brendan McKay wrote:
> This is a sequence simply derived from an existing sequence.
> Thousands of sequences like this could be added but unless
> they have important applications in their own right they
> should NOT be added imo.
> Brendan.

I disagree. One of the most important applications of OEIS now
and in the future will surely be its role as a "fingerprint database",
and we cannot know beforehand know which particular variant
of the sequence the future researchers will encounter,
is it the sequence A, which is say "transformation X" of the sequence B,
or is it B, which is the "transformation X^-1" of the A.

And from the workings of the SuperSeeker I have understand
that in this case it would be even impossible for it
to go from A069623 to A001597.

Also, this one certainly deserves its existence, because of
that beatiful formula, not immediately obvious at all to me.

> > %I A069623
> > %S A069623
> > 1,1,1,2,2,2,2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,8,8,8,8,9,
> > %T A069623
> > 9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
> > %U A069623
> > 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12
> > %N A069623 Number of perfect powers <= n.
> > %F A069623 a(n) = n - SUM(k = 2 to [log2(n)], mu(k) [n^(1/e)-1]), where mu = A008683
> > %Y A069623 Perfect powers are A001597
> >

And one metamathematical, quasiphilosophical point:

I guess that the vast majority of OEIS sequences can be
computed with a primitive recursive function, thus in
a sense, all the sequences are eventually "simply derived" from
the all-zero sequence and the successor function S(n) = n+1
(~ A000027).

I don't think the amount of the sequences is the real
problem of OEIS (except for Neil's work load), but the
organization (or the lack of it).

Furthermore, if Neil is really thinking about adding new
keywords, then I suggest something like
"enum" or "cnts" for the sequences that count something
(and hard-core combinatorialists could limit their searches
with something like +enum and -base as Wouter recently suggested),
"enco" or "code" for binary and other encodings of
more complicated objects, "numt" for number theoretics
related (which are not of "base" nature),
"nper" for the permutations of natural numbers,

(As long as there is no own subsearch field for the
keywords line only, it's good to choose such
four-letter abbreviations that do not normally occur
on the other lines.)


Antti Karttunen

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