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# Harshad numbers

A base **Harshad number** is a positive integer that is divisible by the sum of its base digits. For example, in the decimal numeral system, 1729 is a Harshad number since 1 + 7 + 2 + 9 = 19, and 1729 = 19 × 91. Also called **Niven numbers** or, much less commonly, **multidigital numbers**.

It is important to note that the definition requires divisibility by the sum of digits, not the digital root. One could be led astray to think that all multiples of 9 are Harshad numbers in decimal, but 99 is not: 9 + 9 = 18, which is even, but 99 is odd. In general, many consecutive multiples of can be found to be Harshad numbers in base .

Single digit numbers are trivially Harshad numbers, and in fact we can say that the only way for a positive prime number to be a Harshad number in base is for to hold true.

The following table shows first few Harshad numbers greater than .

Small Harshad numbers | A-number | |

2 | 4, 6, 8, 10, 12, 16, 18, 20, 21, 24, 32, 34, 36, 40, 42, 48 | A049445 |

3 | 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32 | A064150 |

4 | 6, 8, 9, 12, 16, 18, 20, 21, 24, 28, 30, 32, 33, 35, 36, 40, 42 | A064438 |

5 | 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 26, 27, 28, 30, 32, 36 | A064481 |

6 | 10, 12, 15, 18, 20, 24, 25, 30, 36, 40, 42, 44, 45, 48, 50, 55 | |

7 | 8, 9, 12, 14, 15, 16, 18, 21, 24, 27, 28, 30, 32, 35, 36, 40, 42 | |

8 | 14, 16, 21, 24, 28, 32, 35, 40, 42, 48, 49, 56, 64, 66, 70, 72 | A245802 |

9 | 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 45, 48, 50, 54 | |

10 | 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63 | A005349 |

11 | 12, 15, 20, 22, 24, 25, 30, 33, 35, 36, 40, 44, 45, 48, 50, 55 | |

12 | 22, 24, 33, 36, 44, 48, 55, 60, 66, 72, 77, 84, 88, 96, 99, 108 |

Only 1, 2, 4, 6 are Harshad numbers in every integer base .