In recreational mathematics, a harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers.
Examples[]
The number 134 is Harshad number since 134 is divisible by 1+3+4 = 8 (134/8 = 1E, which is integer).
The number 140 is not Harshad number since 140 is not divisible by 1+4+0 = 5 (140/5 = 32.497249724972..., which is not integer) (it can be noted that 140 is the smallest multiple of 10 that is not Harshad number)
The Harshad numbers up to 1000[]
1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 1X, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, X0, X1, E0, 100, 10X, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1X0, 1E0, 1EX, 200, 209, 20E, 210, 216, 218, 220, 223, 227, 22X, 236, 240, 244, 245, 249, 254, 260, 263, 268, 272, 281, 287, 290, 2X0, 2X6, 2XX, 300, 308, 30X, 310, 311, 314, 316, 317, 326, 330, 331, 335, 338, 344, 353, 360, 362, 366, 371, 380, 390, 394, 39X, 3E8, 3EE, 400, 407, 409, 416, 420, 423, 425, 434, 440, 443, 446, 452, 45X, 461, 470, 480, 483, 488, 48X, 4X8, 4E6, 500, 506, 508, 510, 515, 524, 533, 539, 542, 550, 551, 554, 560, 570, 576, 57X, 598, 5X0, 5XX, 5E3, 5E6, 600, 605, 607, 614, 620, 622, 623, 628, 630, 632, 641, 650, 660, 662, 669, 66X, 674, 688, 6X6, 700, 704, 706, 713, 722, 731, 740, 748, 750, 754, 75X, 770, 778, 794, 796, 799, 7E4, 800, 803, 805, 812, 816, 821, 830, 840, 846, 84X, 868, 880, 886, 8X1, 8X4, 8E4, 900, 902, 904, 911, 920, 926, 930, 933, 938, 93X, 958, 976, 988, 990, 993, 994, 9E2, X00, X01, X03, X10, X20, X2X, X36, X48, X50, X61, X66, X84, X85, XX0, XX2, XX8, E00, E02, E10, E14, E16, E1X, E38, E46, E56, E74, E77, E92, EE0, 1000
Properties[]
Smallest k such that k×n is Harshad number are
| +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +X | +E | +10 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 10+ | 10 | 6 | 4 | 3 | X | 2 | E | 3 | 4 | 1 | 7 | 1 |
| 20+ | 10 | 6 | 4 | 3 | E | 2 | E | 3 | 1 | 5 | 9 | 1 |
| 30+ | 10 | E | 4 | 3 | E | 2 | E | 1 | 4 | 4 | E | 1 |
| 40+ | 14 | 6 | 4 | 3 | E | 2 | 1 | 3 | E | E | E | 1 |
| 50+ | 10 | E | 5 | 7 | 9 | 1 | 7 | 3 | 3 | 9 | E | 1 |
| 60+ | 1X | 6 | 4 | E | 1 | 2 | E | 9 | X | E | E | 1 |
| 70+ | 2 | E | 18 | 1 | 5 | 2 | 8 | 2 | E | E | 9 | 1 |
| 80+ | 1X | 8 | 1 | 3 | 1X | 2 | 19 | 3 | 3 | E | E | 1 |
| 90+ | 1X | 1 | 4 | 3 | 1X | E | 2 | E | 4 | E | 4 | 1 |
| X0+ | 1 | 6 | 1X | E | 9 | 4 | 12 | 7 | 1X | 6 | E | 1 |
| E0+ | 8 | 6 | 4 | 3 | 1X | 2 | 1X | 6 | 14 | E | 2 | 1 |
| 100+ | 10 | E | 10 | 3 | 10 | 2 | 1X | 7 | 2 | 1 | 10 | 1 |
| 110+ | 10 | E | E | 7 | 1 | 5 | E | E | 1 | E | E | 1 |
| 120+ | 6 | 1 | 5 | E | E | X | 8 | 1 | X | 5 | E | 1 |
| 130+ | 1X | 4 | 4 | 1 | E | X | 1 | X | 5 | E | E | 3 |
| 140+ | 10 | E | 3 | 4 | E | 1 | 9 | 3 | 3 | E | E | 1 |
| 150+ | 1X | E | 1 | 3 | 1 | 3 | E | E | E | E | E | 2 |
| 160+ | X | E | 8 | 1 | 2 | 2 | E | 2 | 2 | E | E | 7 |
| 170+ | 56 | 1 | 1 | 9 | 1X | 2 | 8 | 9 | 8 | 2 | E | 3 |
| 180+ | 1X | 1 | 8 | 3 | 10 | E | 6 | X | 8 | 6 | E | 2 |
| 190+ | 1 | 7 | 2 | E | 1X | E | 12 | 3 | 8 | E | E | 1 |
| 1X0+ | 10 | 4 | 4 | 3 | 1X | 2 | 1X | 2 | 4 | E | 2 | 1 |
| 1E0+ | 1X | E | X | 3 | 18 | 8 | 1X | E | 3 | 1 | 17 | 1 |
Smallest k such that k×n is not Harshad number are
| +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +X | +E | +10 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0+ | 11 | 7 | 5 | 4 | 3 | 3 | 2 | 2 | 2 | 2 | 11 | 14 |
| 10+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 111 | 1 | 8 |
| 20+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 1 | 1 | 6 |
| 30+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 111 | 1 | 1 | 1 | 4 |
| 40+ | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 1 | 1 | 1 | 1 | 4 |
| 50+ | 1 | 1 | 1 | 1 | 1 | 1111 | 1 | 1 | 1 | 1 | 1 | 3 |
| 60+ | 1 | 1 | 1 | 1 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| 70+ | 1 | 1 | 1 | 82 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| 80+ | 1 | 1 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| 90+ | 1 | 3E | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| X0+ | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11111 |
| E0+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 14 |
| 100+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 37 | 1 | 13 |
| 110+ | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 5 | 1 | 1 | 7 |
| 120+ | 1 | 6 | 1 | 1 | 1 | 1 | 1 | 41 | 1 | 1 | 1 | 6 |
| 130+ | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 |
| 140+ | 1 | 1 | 1 | 1 | 1 | 511 | 1 | 1 | 1 | 1 | 1 | 3 |
| 150+ | 1 | 1 | 3 | 1 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 160+ | 1 | 1 | 1 | 3E | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 170+ | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 180+ | 1 | 2E | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 190+ | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11111 |
| 1X0+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| 1E0+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 101 | 1 | 8 |
Given the divisibility test for E, one might be tempted to generalize that all numbers divisible by E are also harshad numbers. But for the purpose of determining the harshadness of n, the digits of n can only be added up once and n must be divisible by that sum; otherwise, it is not a harshad number. For example, EE is not a harshad number, since E + E = 1X, and EE is not divisible by 1X.
The base number (and furthermore, its powers) will always be a harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1.
All numbers whose base b digit sum divides b−1 are harshad numbers in base b.
For a prime number to also be a harshad number, it must be less than or equal to the base number. Otherwise, the digits of the prime will add up to a number that is more than 1 but less than the prime and, obviously, it will not be divisible. For example: 11 is not harshad in base 10 because the sum of its digits "11" is 1 + 1 = 2, and 11 is not divisible by 2.
The natural density of the harshad numbers is about 0.0925 or 9.25% (there are 92481 harshad numbers ≤ 106).
Although the sequence of factorials starts with Harshad numbers, not all factorials are Harshad numbers, after 7! (=2E00, whose digit sum is 11), the next counterexample is 8X4! (whose digit sum is 8275 = E*8E7, thus not divide 8X4!).
There are no 21 consecutive integers that are all Harshad numbers, but there are infinitely many 20-tuples of consecutive integers that are all Harshad numbers, this is proved by Cooper and Kennedy in 11X1. H. G. Grundman extended the Cooper and Kennedy result to show that there are 2b but not 2b + 1 consecutive b-harshad numbers. This result was strengthened to show that there are infinitely many runs of 2b consecutive b-harshad numbers for b = 2 or 3 by T. Cai and for arbitrary b by Brad Wilson in 11X5.
A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and 6 (The number 10 is a harshad number in all bases except octal (base 8), however, we can note that 10 is the least common multiple of the only four all-harshad numbers).
Nivenmorphic number[]
A Nivenmorphic number or harshadmorphic number for a given number base is an integer t such that there exists some harshad number N whose digit sum is t, and t, written in that base, terminates N written in the same base.
For example, 16 is a Nivenmorphic number for base 10:
7416 is a harshad number
7416 has 16 as digit sum
16 terminates 7416
Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11.
Super harshad numbers[]
A super harshad number (or super Niven number) is a number divisible by the sums of all the nonempty subsets of their nonzero digits, the first few in dozenal are:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 20, 30, 40, 50, 60, 70, 80, 90, X0, E0, 100, 110, 120, 130, 200, 210, 220, 240, 260, 290, 300, 310, 330, 360, 380, 390, 400, 420, 440, 480, 500, 550, 5X0, 600, 620, 630, 660, 700, 770, 800, 840, 880, 900, 930, 990, X00, X50, XX0, E00, EE0, 1000, ...
Not all super harshad numbers ≥10 end with 0, the first few counterexamples are
- 10004, 20008, 1000006, 100000004, 200000008, 1000000000004, 1000000000006, ... (these 7 numbers seem to be the only counterexamples ≤1010+10)
However, all super harshad numbers ≥10 contain at least one digit 0.
A primitive super harshad number (or primitive super Niven number) is a super harshad number n such that, either n does not end with 0 (i.e. n is not divisible by 10), or n/10 is not super harshad number, the first few primitive super harshad numbers are:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 110, 120, 130, 210, 220, 240, 260, 290, 310, 330, 360, 380, 390, 420, 440, 480, 550, 5X0, 620, 630, 660, 770, 840, 880, 930, 990, X50, XX0, EE0, 1010, 1020, 1030, 1110, 1120, 1140, 1190, 1210, 1220, 1800, 2010, 2020, 2040, 2060, 2090, 2110, 2220, 2230, 2240, 2280, 2310, 2360, 2420, 2440, 3010, 3030, 3060, 3080, 3090, 3110, 3310, 3320, 3330, 3360, 3630, 3660, 4020, 4040, 4080, 4090, 4220, 4340, 4440, 4460, 4480, 4620, 4840, 4880, 5050, 50X0, 5550, 55X0, 5X50, 5XX0, 6020, 6030, 6060, 6220, 6330, 6620, 6640, 6660, 6690, 6930, 7070, 7770, 8040, 8080, 8100, 8300, 8440, 8680, 8880, 9030, 9090, 9200, 9330, 9930, 9960, 9990, X050, X0X0, X550, XXX0, E0E0, EEE0, ...
Zuckerman numbers[]
A Zuckerman number (or multiplication-harshad number or multiplication-Niven number) in a given number base, is an integer that is divisible by the product of its digits when written in that base, Zuckerman numbers are hence Nude numbers (number divisible by all of its digits), a Zuckerman numbers cannot contain the zero digit (0), since no nonzero numbers are divisible by 0.
Zuckerman number is a generalization of Harshad number, "Zuckerman number" to "product" is "Harshad number" to "sum".
The Zuckerman numbers are:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 11, 12, 13, 14, 16, 28, 111, 112, 113, 114, 116, 128, 139, 151, 156, 173, 214, 228, 316, 353, 414, 454, 513, 571, 712, 8X8, E56, 1111, 1112, 1113, 1114, 1116, 1128, 1139, 1153, 1174, 11X8, 1214, 1228, 1254, 1316, 1414, 1528, 1713, 1754, 1X28, 1E28, 2114, 2214, 2228, 2714, 3116, 3176, 3739, 4114, 4154, 5114, 5156, 5228, 5316, 5513, 5576, 7139, 8114, 8514, 9116, X114, X228, E128, E254, ...
Sum-product numbers is a subset of both Harshad numbers and Zuckerman numbers, a number n is sum-product number iff n = (sum of digits of n) * (product of digits of n), there are only 5 such numbers: 0, 1, 128, 173, 353
Fiven numbers[]
A Fiven number (or harshad number in factorial base or Niven number in factorial base) is a number which is divisible by the sum of their factorial base digits, the first few Fiven numbers are: (written in dozenal)
- 1, 2, 4, 6, 8, 9, 10, 14, 16, 18, 20, 22, 23, 26, 2E, 30, 34, 40, 44, 46, 48, 50, 5X, 60, 63, 68, 76, 77, 80, 89, 90, 94, 99, X0, X2, X3, X6, E0, E3, E8, 100, 104, 106, 108, 110, 115, 120, 127, 130, 136, 140, 150, 154, 156, 166, 168, 180, 184, 185, 186, 188, 18X, 190, 1X0, 1E4, 1E6, 200, 202, 210, 211, 214, 223, 226, 240, 248, 260, 26E, 276, 280, 281, 286, 290, 294, 299, 2X6, 2E0, 2E3, 2EX, 300, 302, 308, 309, 316, 323, 330, 332, 340, 34X, 350, 354, 366, 367, 380, 388, 39X, 3E6, 3E8, 400, 404, 416, 41X, 420, 426, 440, 446, 448, 454, 480, 488, 48X, 495, 4X8, 4E7, 500, 502, 503, 506, 510, 513, 518, 520, 524, 526, 528, 530, 540, 547, 550, 554, 556, 560, 570, 571, 574, 583, 586, 5X0, 5X4, 5X5, 5X6, 5X8, 5XX, 5E0, 5E9, 600, 60E, 616, 620, 622, 630, 634, 639, 646, 650, 653, 65E, 660, 662, 668, 669, 676, 679, 680, 696, 698, 6X0, 6X1, 6X6, 6XX, 6E0, 6E4, 706, 707, 71X, 720, 728, 74E, 752, 75E, 760, 765, 770, 774, 786, 7X0, 7X6, 7X8, 816, 820, 824, 840, 844, 846, 853, 860, 861, 868, 869, 874, 876, 883, 890, 8X0, 8X6, 8X8, 900, 920, 928, 938, 93X, 940, 958, 960, 970, 976, 99X, 9E0, 9E2, X00, X04, X05, X06, X08, X0X, X10, X20, X36, X38, X40, X42, X4X, X50, X54, X66, X67, X80, X88, XX0, XE3, XE6, E00, E01, E05, E06, E10, E14, E26, E3X, E40, E46, E48, E72, E74, E80, E90, E92, E94, E99, EX4, EX6, EE3, EEE, 1000, ...
Written in factorial base, they are:
- 1, 10, 20, 100, 110, 111, 200, 220, 300, 310, 1000, 1010, 1011, 1100, 1121, 1200, 1220, 2000, 2020, 2100, 2110, 2200, 2320, 3000, 3011, 3110, 3300, 3301, 4000, 4111, 4200, 4220, 4311, 10000, 10010, 10011, 10100, 10200, 10211, 10310, 11000, 11020, 11100, 11110, 11200, 11221, 12000, 12101, 12200, 12300, 13000, 13200, 13220, 13300, 14100, 14110, 20000, 20020, 20021, 20100, 20110, 20120, 20200, 21000, 21220, 21300, 22000, 22010, 22200, 22201, 22220, 23011, 23100, 24000, 24110, 30000, 30121, 30300, 31000, 31001, 31100, 31200, 31220, 31311, 32100, 32200, 32211, 32320, 33000, 33010, 33110, 33111, 33300, 34011, 34200, 34210, 40000, 40120, 40200, 40220, 41100, 41101, 42000, 42110, 42320, 43300, 43310, 44000, 44020, 44300, 44320, 50000, 50100, 51000, 51100, 51110, 51220, 53000, 53110, 53120, 53221, 54110, 54301, 100000, 100010, 100011, 100100, 100200, 100211, 100310, 101000, 101020, 101100, 101110, 101200, 102000, 102101, 102200, 102220, 102300, 103000, 103200, 103201, 103220, 104011, 104100, 110000, 110020, 110021, 110100, 110110, 110120, 110200, 110311, 111000, 111121, 111300, 112000, 112010, 112200, 112220, 112311, 113100, 113200, 113211, 113321, 114000, 114010, 114110, 114111, 114300, 114311, 120000, 120300, 120310, 121000, 121001, 121100, 121120, 121200, 121220, 122100, 122101, 122320, 123000, 123110, 124121, 124210, 124321, 130000, 130021, 130200, 130220, 131100, 132000, 132100, 132110, 133300, 134000, 134020, 140000, 140020, 140100, 140211, 141000, 141001, 141110, 141111, 141220, 141300, 142011, 142200, 143000, 143100, 143110, 144000, 150000, 150110, 150310, 150320, 151000, 151310, 152000, 152200, 152300, 153320, 154200, 154210, 200000, 200020, 200021, 200100, 200110, 200120, 200200, 201000, 201300, 201310, 202000, 202010, 202120, 202200, 202220, 203100, 203101, 204000, 204110, 210000, 210211, 210300, 211000, 211001, 211021, 211100, 211200, 211220, 212100, 212320, 213000, 213100, 213110, 214210, 214220, 220000, 220200, 220210, 220220, 220311, 221020, 221100, 221211, 221321, 222000, ...
There are 187 Niven numbers (in dozenal) ≤1000, while there are 1X9 Fiven numbers ≤1000, the density of Fiven numbers seems to be a few more than the density of Niven numbers (in dozenal).
Mod n[]
Pictorial representation of remainders (mod 1, 2, 3, ...,10) frequency for such numbers ≤ 106:
| modulo | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 92481 | |||||||||||
| 2 | 6E424 | 23059 | ||||||||||
| 3 | 58075 | 17868 | 1X760 | |||||||||
| 4 | 46859 | 1153E | 24787 | 1171X | ||||||||
| 5 | 27990 | 17884 | 176EX | 17725 | 17806 | |||||||
| 6 | 3EEX7 | 3X01 | 17592 | 1808X | 13X67 | 318X | ||||||
| 7 | 1X9E3 | 1272E | 12730 | 1270E | 126XX | 126X4 | 1274X | |||||
| 8 | 27770 | 6X5X | 12614 | 695E | 1E0X9 | 66X1 | 12173 | 697E | ||||
| 9 | 234X3 | 6783 | 7857 | 1829E | 6720 | 7825 | 184E3 | 6585 | 72X0 | |||
| X | 1EE81 | 49X5 | 12994 | 4950 | 12X17 | 7X0E | 12X9E | 4926 | 12995 | 49XE | ||
| E | 48056 | 58E0 | 62E3 | 4447 | 5688 | 5942 | 5024 | 5724 | 5536 | 5332 | 53E7 | |
| 10 | 27514 | 2368 | 8397 | X734 | 1014X | 1839 | 14693 | 1655 | E1E7 | 9556 | 3919 | 1551 |
| remainder | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E |
A list for how many such numbers ≤ 106 are multiples of the numbers from 1 to 100.
| +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +X | +E | +10 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0+ | 92481 | 6E424 | 58075 | 46859 | 27990 | 3EEX7 | 1X9E3 | 27770 | 234X3 | 1EE81 | 48056 | 27514 |
| 10+ | 10498 | 15359 | 1717E | 15E94 | 9713 | 17676 | 8686 | 13X4X | 11840 | 35289 | 704E | 16632 |
| 20+ | 8288 | X203 | XX77 | E857 | 6187 | 12008 | 5E9E | X26X | 30X12 | 7694 | 7584 | 11352 |
| 30+ | 4X65 | 6767 | 713X | 9351 | 3E24 | 9X81 | 3619 | 21045 | 777E | 50X1 | 2975 | X3X2 |
| 40+ | 35X7 | 5X26 | 5199 | 64E8 | 21E0 | 7594 | E742 | 65E9 | 458X | 40X6 | 1X59 | 9120 |
| 50+ | 1998 | 3XE4 | 52E5 | 5147 | 3201 | 1XE9X | 17X9 | 4692 | 3857 | 52E4 | 167E | 7998 |
| 60+ | 1626 | 3204 | 459E | 4014 | 8008 | 5320 | 1463 | 5116 | 3760 | 2730 | 1368 | 6636 |
| 70+ | 2606 | 2478 | 31E0 | 1051E | 12E6 | 5612 | 23X1 | 3155 | 3021 | 1E80 | 22X0 | 5655 |
| 80+ | 1177 | 268X | 10528 | 3898 | 1103 | 3927 | 1100 | 3735 | 4034 | 1722 | 1032 | 4EEE |
| 90+ | 1013 | 84E5 | 2545 | 3698 | E96 | 3328 | 1X18 | 2609 | 2964 | 14X8 | 195E | 5262 |
| X0+ | 5111 | 1458 | 2051 | 249E | 17X6 | 3991 | X61 | 2689 | 1X34 | 2726 | X01 | 11110 |
| E0+ | 1718 | 1305 | 2X86 | 2681 | 970 | 26X8 | 989 | 3494 | 16X2 | 1213 | 4306 | 462E |
External Links[]
- Harshad number on Wikipedia
