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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

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)

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.

Base 10 multiplicative digital root generating function

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][dead link][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

Base 10 multiplicative persistence generating function

Partial sums of base 10 multiplicative digital roots

What about the partial sums

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

and upper bound

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

Base 10 nonzero multiplicative digital root properties

(...)

Base 10 nonzero multiplicative digital root generating function

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

Base 10 nonzero multiplicative persistence generating function

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