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
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A031347:
Multiplicative digital root of
(keep multiplying [base
10] digits of
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
? (Of course it is
, but more specifically?) How often does
? Etc.
Since among the
nonnegative integers in
with
digits [base
10], there are (the first digit being nonzero)
-
k − 1 10 k = 10 ⋅ 9 k − 1, k ≥ 2, |
integers not containing the digit 0, and
-
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
-
which will then grow by a nonzero finite amount (1 to 9) asymptotically 0% of the time.
See also: