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Multiplicative digital root

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The base
b
multiplicative digital root of an integer
n
is the result of repeatedly multiplying the base
b
digits of
n
until reaching a single digit. For example, the base 10 multiplicative digital root of 1729 is 1 × 7 × 2 × 9 = 126, which leads to 1 × 2 × 6 = 12 and then 1 × 2 = 2.

Base 10 multiplicative digital root

A031347 Multiplicative digital root of
n, n   ≥   0
: repeatedly multiply digits until reaching a single digit, the multiplicative digital root.
{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, ...}

Numbers having multiplicative digital root
n

n
Sequence A-number
0 {0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, 59, 60, 65, 69, 70, 78, 80, 85, 87, 90, 95, 96, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 125, 130, 140, 145, ...} A034048
1 {1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, ...} A002275
 (n), n   ≥   1
2 {2, 12, 21, 26, 34, 37, 43, 62, 73, 112, 121, 126, 134, 137, 143, 162, 173, 211, 216, 223, 232, 261, 278, 279, 287, 297, 299, 314, 317, 322, 341, 367, 369, 371, 376, 389, 396, 398, ...} A034049
3 {3, 13, 31, 113, 131, 311, 1113, 1131, 1311, 3111, 11113, 11131, 11311, 13111, 31111, 111113, 111131, 111311, 113111, 131111, 311111, 1111113, 1111131, 1111311, 1113111, ...} A034050
4 {4, 14, 22, 27, 39, 41, 72, 89, 93, 98, 114, 122, 127, 139, 141, 172, 189, 193, 198, 212, 217, 221, 249, 266, 271, 277, 294, 319, 333, 338, 346, 364, 379, 383, 391, 397, 411, 429, ...} A034051
5 {5, 15, 35, 51, 53, 57, 75, 115, 135, 151, 153, 157, 175, 315, 351, 355, 359, 395, 511, 513, 517, 531, 535, 539, 553, 557, 571, 575, 579, 593, 597, 715, 751, 755, 759, 795, 935, 953, ...} A034052
6 {6, 16, 23, 28, 32, 44, 47, 48, 61, 68, 74, 82, 84, 86, 116, 123, 128, 132, 144, 147, 148, 161, 168, 174, 182, 184, 186, 213, 218, 224, 227, 228, 231, 238, 242, 244, 246, 264, 267, ...} A034053
7 {7, 17, 71, 117, 171, 711, 1117, 1171, 1711, 7111, 11117, 11171, 11711, 17111, 71111, 111117, 111171, 111711, 117111, 171111, 711111, 1111117, 1111171, 1111711, 1117111, ...} A034054
8 {8, 18, 24, 29, 36, 38, 42, 46, 49, 63, 64, 66, 67, 76, 77, 79, 81, 83, 88, 92, 94, 97, 99, 118, 124, 129, 136, 138, 142, 146, 149, 163, 164, 166, 167, 176, 177, 179, 181, 183, 188, ...} A034055
9 {9, 19, 33, 91, 119, 133, 191, 313, 331, 911, 1119, 1133, 1191, 1313, 1331, 1911, 3113, 3131, 3311, 9111, 11119, 11133, 11191, 11313, 11331, 11911, 13113, 13131, 13311, 19111, ...} A034056

Base 10 multiplicative digital root formulae

mdr10(n)  =  ?, n ≥ 0.

Base 10 multiplicative digital root properties

(...)

Base 10 multiplicative digital root asymptotic properties

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.

Base 10 multiplicative digital root generating function

G{mdr10(n)}(x)  :=
n  = 0
  
mdr10(n) xn  =  ?

Base 10 multiplicative persistence

A031346 Multiplicative persistence of
n, n   ≥   0
: number of products of digits needed to obtain a single digit (the multiplicative digital root).
{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

mp10(n)  =  ?, n ≥ 0.

Base 10 multiplicative persistence generating function

G{mp10(n)}(x)  :=
n  = 0
  
mp10(n) xn  =  ?

Partial sums of base 10 multiplicative digital roots

What about the partial sums

An:=
n
i  = 0
  
mp10(i )  =  ?,

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

For
n = 10k  −  1
, we have lower bound
10k − 1
i  = 0
  
mp10(i )  >  45 + 1 ⋅  10
k
i  = 2
  
9k  − 1, k ≥ 2,

and upper bound

10k − 1
i  = 0
  
mp10(i )  <  45 + 9  ⋅  10
k
i  = 2
  
9k  − 1, k ≥ 2,

Base 10 nonzero multiplicative digital root

A051802 Nonzero multiplicative digital root of
n, n   ≥   0
: repeatedly multiply nonzero digits until reaching a single digit, the nonzero multiplicative digital root. (For 0 we have the empty product, giving 1.)
{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, ...}

Numbers having nonzero multiplicative digital root
n

n
Sequence A-number
1 {1, ...} A??????
2 {2, ...} A??????
3 {3, ...} A??????
4 {4, ...} A??????
5 {5, ...} A??????
6 {6, ...} A??????
7 {7, ...} A??????
8 {8, ...} A??????
9 {9, ...} A??????

Base 10 nonzero multiplicative digital root formulae

nzmdr10(n)  =  ?, n ≥ 0.

Base 10 nonzero multiplicative digital root properties

(...)

Base 10 nonzero multiplicative digital root generating function

G{nzmdr10(n)}(x)  :=
n  = 0
  
nzmdr10(n) xn  =  ?

Base 10 nonzero multiplicative persistence

A?????? Nonzero multiplicative persistence of
n, n   ≥   0
: number of products of nonzero digits needed to obtain a single digit (the nonzero multiplicative digital root).
{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

nzmp10(n)  =  ?, n ≥ 0.

Base 10 nonzero multiplicative persistence generating function

G{nzmp10(n)}(x)  :=
n  = 0
  
nzmp10(n) xn  =  ?

See also

Notes

  1. 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]
  2. N. J. A. Sloane, “The persistence of a number,” J. Recreational Math, 6 (1973), pp. 97–98.