Modifications for A069623, added formula

David Wilson davidwwilson at
Sat Oct 12 16:13:08 CEST 2002

----- Original Message -----
From: "Brendan McKay" <bdm at>
To: "David Wilson" <davidwwilson at>
Cc: "Sequence Fanatics" <seqfan at>
Sent: Friday, October 11, 2002 9:39 PM
Subject: Re: Modifications for A069623, added formula

> 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.
> * David Wilson <davidwwilson at> [021012 05:18]:
> > %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
> > 9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,
> > %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 = 1 to [log2(n)], mu(k) [n^(1/e)-1]), where
mu =
> > A008683
> > %Y A069623 Perfect powers are A001597.

I don't think derivative sequences should be rejected out of hand.  On that
basis alone, we should reject, say, A000720 (the prime counting function)
on the grounds that it is a derivative of A000040 (the prime numbers),
we know that A000720 is a much-studied function in its own right.

I think a similar argument might be made for the perfect power counting
function A069623.  Though obviously not as important as the prime counting
function, the subject of counting perfect powers periodically arises, as it
did recently in math-fun/seqfan.  This discussion prompted me to add an
efficient formula for A069623, which by itself may justify the presence of
the sequence in the OEIS.

If you are saying "A069623 is an example of garbage, let's not add any more
of this to the database", then I applaud your attitude, though I do not
with your assessment of this particular sequence.  We the editors should
reject more garbage at the "new" phase, I think NJAS has been nudging us
in this direction by lately rejecting several new sequences himself.

If you are saying "A069623 is an example of garbage, let's expunge it from
the database", I say no, because it would be at once a departure from
current policy and ineffective.  A069623 is no longer a "new" sequence.
This, in theory, means that it was reviewed and accepted.  NJAS retains
sole authority to reject admitted sequences, he rarely does so unless the
sequence is short, a duplicate, or has unrectifiable errors.  This is
admitted sequences are potentially referenced in outside publications.
Under this policy, A069623 stays in the database.  Besides, rejecting one
or two admitted sequences is ineffective.  There are literally thousands of
poorly/vaguely defined, inaccurate, and uninteresting sequences in the
database which must be culled to have any discernible impact on OEIS

If you are saying "A069623 is an example of garbage, let's fix the
well, join the chorus.  With all due respect to NJAS, his past open-door
policy on new sequences has caused the OEIS to fall short of the stated goal
of publication-quality material.  If OEIS quality is ever going to improve
significantly, there must be a concerted effort at OEIS reform, including
review and rejection of admitted sequences.  This is significant work, and I
do not believe it will ever happen.  If it does not, rejection of individual
admitted sequences on the basis of irrelevance is unworthy of consideration.

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