[seqfan] Re: A079277 review
David Wilson
davidwwilson at comcast.net
Sun Feb 26 22:04:17 CET 2012
I am reviewing A079277 in relation to some other sequences I added not
long ago.
There is a monolithic comment which I have broken into its five
constituent pieces:
*[0)] The function a(n) complements Eulers phi-function:*
The meaning of this comment is unclear, perhaps it was meant to be
clarified and supported by subsequent comments involving phi, but I
don't think they do, as I note later. I vote to elide this comment.
*1) a(n)+phi(n)=n if n is a power of a prime.*
Technically true, but phi(n) takes on a special form on prime powers:
phi(p^e) = (p-1)*p^(e-1), so this really just says a(p^e) = p^(e-1) in
an obfuscatory fashion. Perhaps we should replace this comment with
"a(p^e) = p^(e-1) for prime p and e >= 1". The relationship between a(n)
and phi(n) on prime powers n says little about a relationship on general
n, so this statement doesn't really support comment 0.
*2) It seems also that a(n)+phi(n)>=n for "almost all numbers".*
This comment is vague. If I take it to mean n satisfying a(n) + phi(n)
>= n have density 1 on the positive integers, I cannot prove it.
However, I can find an (almost certainly) infinite number of
counterexamples, including any number n = pq with primes p, q satisfying
p^2-p+1 < q < p^2, the smallest being n = 115. These counterexamples
take the wind out of comment 0. This comment scores low on clarity,
truth, and relevance, and I vote to elide it.
*3) a(2n)=n+1 if and only if n is a Mersenne prime.*
*4) Lim a(n^k)/n^k =1 if n has at least two prime factors and k goes to
infinity.*
These comments are true and should remain (on separate lines).
Agree? Disagree? Care?
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