[seqfan] Re: A079277 review

[israel at math.ubc.ca]

If I haven't made a mistake, the examples of a(n) + phi(n) < n up to
n=20000 are
115, 329, 1243, 2119, 2171, 4709, 4777, 4811, 6593, 6631, 6707, 6821,
11707, 11983, 12029, 14597, 15463, 16793
Perhaps this is worthy of a sequence of its own.
All these examples have two prime divisors; the only one that is
not the product of two primes is 15463 = 7*47^2.  

Robert Israel
University of British Columbia


On Feb 26 2012, David Wilson wrote:

>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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