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A number    is a Harshad (also called Niven) number if it is divisible by the sum of its digits.

For example, 666 is divisible by 6+6+6 so it is a Harshad number.

C.Cooper & R.E.Kennedy proved that it is possible to have at most 20 consecutive Niven numbers and this happens infinitely often.

L Starts of runs of length L
2 20, 80, 132, 152, 200, 209, 224, 399, 407, 440
3 110, 1010, 2464, 4912, 5054, 5831, 7360, 8203
4 510, 1014, 2022, 3030, 10307, 12102, 12255
5 131052, 491424, 1275140, 1310412, 1474224
6 12751220, 14250624, 22314620, 22604423
7 10000095, 41441420, 207207020, 233735070
8 2162049150, 10810245630, 18039122682
9 124324220, 621621020, 1243242020
10 1, 602102100620, 1002301700420
11 920067411130599
12 43494229746440272890
13 12100324200007455010742303399999999999999999990

Below, the spiral pattern of Harshad numbers up to  . See the page on prime numbers for an explanation and links to similar pictures.

The first Harshad numbers are 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 more terms

Harshad numbers can also be... (you may click on names or numbers and on + to get more values)