until reaching a single digit. For example, the base
.
: 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, ...}
Numbers having multiplicative digital root
|
Sequence
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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
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{1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, ...}
|
A002275
|
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, ...}
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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, ...}
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A034050
|
4
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{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, ...}
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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, ...}
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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, ...}
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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, ...}
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A034055
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9
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{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
|
: 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, ...}
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.
: repeatedly multiply nonzero digits until reaching a single digit, the nonzero multiplicative digital root. (For
-
{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, ...}
: number of products of nonzero digits needed to obtain a single digit (the nonzero multiplicative digital root).