Repdigit numbers

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A (base 10) repdigit number is a number like

d, dd, ddd, dddd, ...

that contains only (zero or more times) the digit

d

, with

0 < d < 10

(in its base 10 representation). The term stands for repeated digit.

Repdigit numbers (in base 10) are numbers of the form

Dn  :=  d  R n  =  d  10 n − 1 10 − 1  =  d   n  − 1 ∑   i  = 0    (10 − 1) 10 i, n ≥ 0, 0 < d < 10,

where

R n

is a repunit number and

D0

is the 0th repdigit, taken as the empty sum (defined as the additive identity, i.e. 0), and thus represented by a string of zero

d

 ’s (the empty string) which is more conveniently written as 0 (the only case where a leading zero is shown).

A010785 Repdigit numbers, or numbers with repeated digits.

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, ...}

A (base 10) repdigit prime must be a (base 10) repunit prime.

Generalized repdigit numbers

A (base

b

) generalized repdigit number is a number like

d, dd, ddd, dddd, ...

that contains only (zero or more times) the digit

d

, with

0 < d < b

(in its base

b

representation). Generalized repdigit numbers (in base

b

) are numbers of the form

D  (b) n  :=  d  R  (b)  n  =  d  b n − 1 b − 1  =  d   n  − 1 ∑   i  = 0    (b − 1) b i, b ≥ 2, n ≥ 0, 0 < d < b,

where

R  (b)  n

is a generalized repunit number and

D (b) 0

is the 0th generalized repunit (base

b

), taken as the empty sum (defined as the additive identity, i.e. 0), and thus represented by a string of zero

d

 ’s (the empty string) which is more conveniently written as 0 (the only case where a leading zero is shown). A (base

b

) generalized repdigit prime must be a (base

b

) generalized repunit prime.

Generalized repdigit numbers (base

b

) converted to base 10

b	Generalized repdigit numbers (base b )	A-number
2	{0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, ...}	A000225
3	{0, 1, 2, 4, 8, 13, 26, 40, 80, 121, 242, 364, 728, 1093, 2186, 3280, 6560, 9841, 19682, 29524, 59048, 88573, 177146, 265720, 531440, 797161, 1594322, 2391484, 4782968, 7174453, ...}	A048328
4	{0, 1, 2, 3, 5, 10, 15, 21, 42, 63, 85, 170, 255, 341, 682, 1023, 1365, 2730, 4095, 5461, 10922, 16383, 21845, 43690, 65535, 87381, 174762, 262143, 349525, 699050, 1048575, 1398101, ...}	A048329
5	{0, 1, 2, 3, 4, 6, 12, 18, 24, 31, 62, 93, 124, 156, 312, 468, 624, 781, 1562, 2343, 3124, 3906, 7812, 11718, 15624, 19531, 39062, 58593, 78124, 97656, 195312, 292968, 390624, 488281, ...}	A048330
6	{0, 1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 43, 86, 129, 172, 215, 259, 518, 777, 1036, 1295, 1555, 3110, 4665, 6220, 7775, 9331, 18662, 27993, 37324, 46655, 55987, 111974, 167961, 223948, ...}	A048331
7	{0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 57, 114, 171, 228, 285, 342, 400, 800, 1200, 1600, 2000, 2400, 2801, 5602, 8403, 11204, 14005, 16806, 19608, 39216, 58824, 78432, 98040, 117648, ...}	A048332
8	{0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 73, 146, 219, 292, 365, 438, 511, 585, 1170, 1755, 2340, 2925, 3510, 4095, 4681, 9362, 14043, 18724, 23405, 28086, 32767, 37449, 74898, ...}	A048333
9	{0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 91, 182, 273, 364, 455, 546, 637, 728, 820, 1640, 2460, 3280, 4100, 4920, 5740, 6560, 7381, 14762, 22143, 29524, 36905, 44286, ...}	A048334
10	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, ...}	A010785
11	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 133, 266, 399, 532, 665, 798, 931, 1064, 1197, 1330, 1464, 2928, 4392, 5856, 7320, 8784, 10248, 11712, 13176, ...}	A048335
12	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 157, 314, 471, 628, 785, 942, 1099, 1256, 1413, 1570, 1727, 1885, 3770, 5655, 7540, 9425, 11310, 13195, ...}	A048336
13	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 183, 366, 549, 732, 915, 1098, 1281, 1464, 1647, 1830, 2013, 2196, 2380, 4760, 7140, 9520, 11900, ...}	A048337
14	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 211, 422, 633, 844, 1055, 1266, 1477, 1688, 1899, 2110, 2321, 2532, 2743, 2955, 5910, ...}	A048338
15	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 241, 482, 723, 964, 1205, 1446, 1687, 1928, 2169, 2410, 2651, 2892, 3133, 3374, ...}	A048339
16	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 273, 546, 819, 1092, 1365, 1638, 1911, 2184, 2457, 2730, 3003, 3276, ...}	A048340
17	{0, 1, ...}	A??????
18	{0, 1, ...}	A??????
19	{0, 1, ...}	A??????
20	{0, 1, ...}	A??????

Recurrence

D  (b) n  =  ?, D  (b) 0  =  0, D  (b) 1  =  1,

where

D  (b) 0

is the 0th generalized repdigit (base

b

), taken as the empty sum (defined as the additive identity, i.e. 0), and thus represented by a string of zero

d

 ’s (the empty string) which is more conveniently written as 0 (the only case where a leading zero is shown).

Generating function

The o.g.f. for

D  (b) n

is

G{D  (b) n}(x)  :=    ∞ ∑   n  = 0    D  (b) n x n  =  ?.

Sequences

A?????? Numbers which can be represented as a string of three or more repeated digits

d, 0 < d < b

in a base

b   ≥   2

.

{?, ...}

See also

• Repunit numbers