Moebius-Function-Based Sequences

[Robert G. Wilson v]

	For reference sake only or in case you wish to submit:
6, 3, 5, 8, 7, 10, 11, 9, 12, 14, 13, 16, 17, 15, 21, 18, 19, 20, 23, 24, 22, \
26, 29, 25, 27, 33, 28, 32, 30, 31, 37, 36, 34, 35, 38, 40, 41, 39, 46, 44, \
42, 43, 47, 45, 48, 51, 53, 49, 50, 52, 55, 54, 59, 56, 57, 60, 58, 62, 61, \
63, 66, 65, 64, 68, 69, 67, 70, 72, 74, 71, 73, 75, 78, 77, 76, 80, 82, 79, \
83, 81, 84, 85, 89, 88, 86, 87, 91, 90, 97, 92, 93, 96, 94, 95, 106, 98, 101, \
99, 100, 104


	What I found more intriguing was the sequence where b(k) = a(k)-k first
equals n (limit 10000000):
2, 3, 9, 4, 1, 15, 26, 89, 146, 139, 95, 199, 114, 293, 1131, 2163, 7393, \
13173, 6559, 2681, 14545, 71719, 97359, 38399, 157474, 276815, 81543, 246405, \
426854, 41223, 544322, 3339078, 1493362, 2156847, 7793502, 293035, 8812153

	Also Sum_{1..10^m) b(n): 30, 304, 3001, 30000, 300001, 3000000, 30000003. So to 
answer your question about a limit, at least using the data above, the answer is 
affirmative and it is 3.


Leroy Quet wrote:
> "mu(k)" is the Moebius (Mobius) function,
> defined by:
> 
> sum{k=1 to oo} mu(k)/k^r  = 1/zeta(r).
> 
> http://www.research.att.com/projects/OEIS?Anum=A008683
> 
> 
> I do not believe this (interesting, in my opinion) sequence is in the EIS 
> yet:
> 
> a(n) = smallest integer > n where
> mu(n) = mu(a(n)).
> 
> 6, 3, 5, 8, 7, 10, 11, 9, 12, 14, 13, 16, ...
> 
> -
> 
> Also, a related sequence:
> 
> b(n) = a(n) - n:
> 
> 5, 1, 2, 4, 2, 4, 4, 1, 3, 4, 2, 4,...
> 
> In regards to {b(n)}, do the terms of this sequence approach a finite 
> average?
> 
> ie. does x =
> limit{m->oo} (1/m) sum{n=1 to m} b(n)
> 
> exist?
> 
> Does x have a closed form?
> 
> --
> 
> Now, the sequences of n's where
> mu(n) = m(n+1) IS in the EIS.
> 
> http://www.research.att.com/projects/OEIS?Anum=A064148
> 
> But the actual values of mu(c(n)), where c(n) is the n_th term of 
> A064148, is not in the EIS.
> 
> -1, 0, 1, 1, 0, 0, -1, -1 ,...
> 
> --
> 
> Finally, I do not believe the following is in the EIS either:
> 
> d(n) = Number of positive divisors k of n, where
> 
> mu(k) = 1 and  mu(n/k) = -1.
> 
> 0, 1, 1, 0, 1, 0, 1, 0, 0, 0,...
> 
> I get the relation (hopefully correct):
> 
> 4*d(n) + sum{k|n} mu(k)*mu(n/k) =
> 
> product{p|n} e(p,n),
> 
> where the product is over the distinct primes dividing n;
> e(p,n) = 2 if p|n but p^2 does not divide n;
> e(p,n) = 1 if p^2|n but p^3 does not divide n;
> e(p,n) = 0 if p^3|n.
> 
> thanks,
> Leroy Quet
>