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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 Edit

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 Edit

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 Edit

Smallest k such that k×n is Harshad number are

+1 +2 +3 +4 +5 +6 +7 +8 +9 +X +E +10
0+ 111111111111
10+ 10643X2E34171
20+ 10643E2E31591
30+ 10E43E2E144E1
40+ 14643E213EEE1
50+ 10E57917339E1
60+ 1X64E12E9XEE1
70+ 2E1815282EE91
80+ 1X8131X21933EE1
90+ 1X1431XE2E4E41
X0+ 161XE941271X6E1
E0+ 86431X21X614E21
100+ 10E1031021X721101
110+ 10EE715EE1EE1
120+ 615EEX81X5E1
130+ 1X441EX1X5EE3
140+ 10E34E1933EE1
150+ 1XE1313EEEEE2
160+ XE8122E22EE7
170+ 561191X28982E3
180+ 1X18310E6X86E2
190+ 172E1XE1238EE1
1X0+ 104431X21X24E21
1E0+ 1XEX31881XE31171

Smallest k such that k×n is not Harshad number are

+1 +2 +3 +4 +5 +6 +7 +8 +9 +X +E +10
0+ 117543322221114
10+ 11111111111118
20+ 1111111111116
30+ 11111111111114
40+ 111111511114
50+ 111111111111113
60+ 111171111113
70+ 1118211111112
80+ 117111111112
90+ 13E1111111112
X0+ 7111111111111111
E0+ 1111111111114
100+ 11111111137113
110+ 111131115117
120+ 1611111411116
130+ 111311311111
140+ 11111511111113
150+ 113171111111
160+ 1113E11111111
170+ 123111111111
180+ 12E1111111111
190+ 5111111111111111
1X0+ 111111111112
1E0+ 11111111110118

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 are all Harshad numbers, but there are infinitely many 20-tuples of consecutive integers that are all Harshad numbers.

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 Edit

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.