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Template:Sequence of the Day for November 13

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Intended for: November 13, 2012

Timetable

  • First draft entered by ? on August? 13?, 2012
  • Draft reviewed by Daniel Forgues on November 13, 2012
  • Draft to be approved by October 13, 2012
Yesterday's SOTD * Tomorrow's SOTD

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A031347: Multiplicative digital root of
n
(keep multiplying [base 10] digits of
n
until reaching a single digit).
{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, ... }
This is one of the original base-dependent integer sequences in the OEIS, but not much seems to be known about it (at least, judging from its entry). What is the asymptotic behavior of
ai
? (Of course it is
Θ (1)
, but more specifically?) How often does
an = an +1 > 0 
? Etc. Since among the
10k
nonnegative integers in
[ 0, 10k  −  1]
with
k, k   ≥   2,
digits [base 10], there are (the first digit being nonzero)
9
10
k  − 1 10k  =  10  ⋅  9k  − 1, k ≥ 2,

integers not containing the digit 0, and

9
10
k  − 1  =  0,

this implies that, asymptotically, 100% of the multiplicative digital roots are 0, i.e. the asymptotic density of nonzero multiplicative digital roots is 0.

What about the partial sums

An:=
n
i  = 1
  
ai  =  ?,

which will then grow by a nonzero finite amount (1 to 9) asymptotically 0% of the time.

See also: