Harshad Number

A Harshad (or Niven) number is a natural number divisible by the sum of its own digits.

The term Harshad comes from Sanskrit and means that which brings joy.

Take a number $ N $ and calculate the sum of all its digits.

Then check if the division of the number by this sum is an integer (i.e., the remainder is zero).

If so, the number $ N $ is a Harshad number.

The list of Harshad numbers up to 1000 is: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204, 207, 209, 210, 216, 220, 222, 224, 225, 228, 230, 234, 240, 243, 247, 252, 261, 264, 266, 270, 280, 285, 288, 300, 306, 308, 312, 315, 320, 322, 324, 330, 333, 336, 342, 351, 360, 364, 370, 372, 375, 378, 392, 396, 399, 400, 402, 405, 407, 408, 410, 414, 420, 423, 432, 440, 441, 444, 448, 450, 460, 465, 468, 476, 480, 481, 486, 500, 504, 506, 510, 511, 512, 513, 516, 518, 522, 531, 540, 550, 552, 555, 558, 576, 588, 592, 594, 600, 603, 605, 612, 621, 624, 629, 630, 640, 644, 645, 648, 660, 666, 684, 690, 700, 702, 704, 711, 715, 720, 730, 732, 735, 736, 738, 756, 770, 774, 777, 780, 782, 792, 800, 801, 803, 804, 810, 820, 825, 828, 832, 840, 846, 864, 870, 874, 880, 882, 888, 900, 902, 910, 912, 915, 918, 935, 936, 954, 960, 966, 972, 990, 999, 1000

See the sequence OEIS A005349 here

Yes. It has been shown that there are infinitely many Harshad numbers.

A Harshad prime number must have a divisor that is either 1 or itself as the sum of its digits.

For the sum of its digits to equal 1, the number must consist of 1s followed by zeros, making it a multiple of 10, and therefore not a prime number.

For the sum of its digits to equal itself, the number must consist of only one digit (itself).

Therefore, the only Harshad prime numbers are the single-digit primes: 2, 3, 5, and 7.

No, not all factorials are Harshad numbers.

Some small factorials, like 3! = 6 or 4! = 24, are Harshad numbers because they are divisible by the sum of their digits.

But 432! is the first factorial that is not a Harshad number.

For each integer, calculate the sum of its digits.

Test if dividing the integer by this sum results in a remainder of zero; if so, it is a Harshad number.

Here is some pseudocode.function harshad(n) {

s ← 0

x ← n

while x > 0 {

s ← s + (x % 10)

x ← x div 10

}

return (n % s = 0)

}