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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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

10, 6, 4, 3, X, 2, E, 3, 4, 1, 7, 1,

10, 6, 4, 3, E, 2, E, 3, 1, 5, 9, 1,

10, E, 4, 3, E, 2, E, 1, 4, 4, E, 1,

14, 6, 4, 3, E, 2, 1, 3, E, E, E, 1,

10, E, 5, 7, 9, 1, 7, 3, 3, 9, E, 1,

1X, 6, 4, E, 1, 2, E, 9, X, E, E, 1,

2, E, 18, 1, 5, 2, 8, 2, E, E, 9, 1,

1X, 8, 1, 3, 1X, 2, 19, 3, 3, E, E, 1,

1X, 1, 4, 3, 1X, E, 2, E, 4, E, 4, 1,

1, 6, 1X, E, 9, 4, 12, 7, 1X, 6, E, 1,

8, 6, 4, 3, 1X, 2, 1X, 6, 14, E, 2, 1,

10, E, 10, 3, 10, 2, 1X, 7, 2, 1, 10, 1,

10, E, E, 7, 1, 5, E, E, 1, E, E, 1,

6, 1, 5, E, E, X, 8, 1, X, 5, E, 1,

1X, 4, 4, 1, E, X, 1, X, 5, E, E, 3,

10, E, 3, 4, E, 1, 9, 3, 3, E, E, 1,

1X, E, 1, 3, 1, 3, E, E, E, E, E, 2,

X, E, 8, 1, 2, 2, E, 2, 2, E, E, 7,

56, 1, 1, 9, 1X, 2, 8, 9, 8, 2, E, 3,

1X, 1, 8, 3, 10, E, 6, X, 8, 6, E, 2,

1, 7, 2, E, 1X, E, 12, 3, 8, E, E, 1,

10, 4, 4, 3, 1X, 2, 1X, 2, 4, E, 2, 1,

1X, E, X, 3, 18, 8, 1X, E, 3, 1, 17, 1,

...

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

11, 7, 5, 4, 3, 3, 2, 2, 2, 2, 11, 14,

1, 1, 1, 1, 1, 1, 1, 1, 1, 111, 1, 8,

1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 6,

1, 1, 1, 1, 1, 1, 1, 111, 1, 1, 1, 4,

1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 4,

1, 1, 1, 1, 1, 1111, 1, 1, 1, 1, 1, 3,

1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 3,

1, 1, 1, 82, 1, 1, 1, 1, 1, 1, 1, 2,

1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2,

1, 3E, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,

7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11111,

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14,

1, 1, 1, 1, 1, 1, 1, 1, 1, 37, 1, 13,

1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 7,

1, 6, 1, 1, 1, 1, 1, 41, 1, 1, 1, 6,

1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1,

1, 1, 1, 1, 1, 511, 1, 1, 1, 1, 1, 3,

1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 1, 1,

1, 1, 1, 3E, 1, 1, 1, 1, 1, 1, 1, 1,

1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1,

1, 2E, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11111,

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,

1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 1, 8,

...

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.

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.