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Multiplicative digital root
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(Redirected from Nonzero multiplicative digital root)
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Contents
- 1 Base 10 multiplicative digital root
- 2 Base 10 nonzero multiplicative digital root
- 3 See also
- 4 Notes
Base 10 multiplicative digital root
A031347 Multiplicative digital root ofn, n ≥ 0 |
- {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, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0, 4, 8, 2, 6, 0, 8, 6, 6, 8, 0, 5, 0, 5, 0, 0, 0, 5, 0, 0, 0, 6, 2, 8, 8, 0, 8, 8, 6, 0, 0, 7, 4, 2, 6, 5, 8, 8, ...}
A?????? First differences of multiplicative digital roots (A031347).
- {1, 1, 1, 1, 1, 1, 1, 1, 1, − 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, − 9, 2, 2, 2, 2, −8, 2, 2, 2, 2, −8, 3, 3, 3, −7, 3, 3, − 6, 6, − 4, − 4, 4, 4, − 6, 4, − 6, 8, −2, 0, ...}
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(n), n ≥ 1 |
Base 10 multiplicative digital root formulae
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mdr10(n) = ?, n ≥ 0.
Base 10 multiplicative digital root properties
(...)
Base 10 multiplicative digital root asymptotic properties
Since among the10 k |
[0, 10 k − 1] |
k, k ≥ 2, |
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k − 1 10 k = 10 ⋅ 9 k − 1, k ≥ 2,9 10
integers not containing the digit 0, and
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k − 1 = 0,9 10
this implies that, asymptotically, 100% of the multiplicative digital roots are 0, i.e. the asymptotic density of nonzero multiplicative digital roots is 0.
Base 10 multiplicative digital root generating function
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G{mdr10(n)}(x) := ∞∑ n = 0
Base 10 multiplicative persistence
A031346 Multiplicative persistence ofn, n ≥ 0 |
- {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 1, 2, 2, 3, 3, 2, 4, ...}
Sloane’s conjecture on multiplicative digital root
In 1973, Neil Sloane conjectured that, with fixed radix bases, no integer has a multiplicative persistence greater than itself.[1][2] So from the earlier example, 1729 is 1 × 7 × 2 × 9 = 126 = 1 × 2 × 6 = 12 = 1 × 2 = 2. The multiplicative persistence is 3 < 1729.
Base 10 multiplicative persistence formulae
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mp10(n) = ?, n ≥ 0.
Base 10 multiplicative persistence generating function
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G{mp10(n)}(x) := ∞∑ n = 0
Partial sums of base 10 multiplicative digital roots
What about the partial sums
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An := n∑ i = 0
which will then grow by a nonzero finite amount (1 to 9) asymptotically 0% of the time.
Forn = 10 k − 1 |
- 10 k − 1
∑ i = 0k∑ i = 2
and upper bound
- 10 k − 1
∑ i = 0k∑ i = 2
Base 10 nonzero multiplicative digital root
A051802 Nonzero multiplicative digital root ofn, n ≥ 0 |
- {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 4, 6, 8, 1, 2, 4, 6, 8, 3, 3, 6, 9, 2, 5, 8, 2, 8, 4, 4, 4, 8, 2, 6, 2, 8, 6, 6, 8, 5, 5, 1, 5, 2, 1, 3, 5, 4, 2, 6, 6, 2, 8, 8, 3, 8, 8, 6, 2, 7, 7, 4, 2, 6, 5, 8, 8, ...}
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Base 10 nonzero multiplicative digital root formulae
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nzmdr10(n) = ?, n ≥ 0.
Base 10 nonzero multiplicative digital root properties
(...)
Base 10 nonzero multiplicative digital root generating function
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G{nzmdr10(n)}(x) := ∞∑ n = 0
Base 10 nonzero multiplicative persistence
A?????? Nonzero multiplicative persistence ofn, n ≥ 0 |
- {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
Base 10 nonzero multiplicative persistence formulae
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nzmp10(n) = ?, n ≥ 0.
Base 10 nonzero multiplicative persistence generating function
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G{nzmp10(n)}(x) := ∞∑ n = 0
See also
- Additive digital root (digital root)
- Multiplicative digital root
- Nonzero multiplicative digital root
Notes
- ↑ L. H., Wilfredo Lopez. “Sloane’s conjecture on multiplicative digital root” (version 4). PlanetMath.org. Freely available at http://planetmath.org/SloanesConjectureOnMultiplicativeDigitalRoot.html[dead link]
- ↑ N. J. A. Sloane, “The persistence of a number,” J. Recreational Math, 6 (1973), pp. 97–98.