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Harshad numbers
A number  $n$  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  $100^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of Harshad numbers

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

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of Harshad numbers

A graph displaying how many Harshad numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Harshad numbers
Useful links Mathworld, Harshad Number
Wikipedia, Harshad number
OEIS, Sequence A005349
Harshad numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 18 24 50 + 9996980000 aban 10 12 18 + 10000000000 abundant 12 18 20 + 50000000 Achilles 72 108 200 + 9999898832 admirable 12 20 24 + 99894896 alt.fact. 5784634181 alternating 10 12 18 + 989898909 amenable 12 20 21 + 1000000000 amicable 220 2620 2924 + 9844469775 anti-perfect 285 apocalyptic 192 220 222 + 30000 arithmetic 20 21 27 + 9999990 astonishing 27 204 216 + 4278092598 automorphic 12890625 787109376 Bell 4140 betrothed 48 140 195 + 9629379500 binomial 10 20 21 + 9999171820 brilliant 10 21 209 + 1679 c.decagonal 3781 20801 50501 + 9715011601 c.heptagonal 736 19426 36058 + 9998597533 c.nonagonal 10 190 820 + 9999171820 c.octagonal 81 225 441 + 9995000625 c.pentagonal 2176 2976 9456 + 9996821151 c.square 481 20605 26221 + 9968014013 c.triangular 10 460 2584 + 9973768960 cake 42 378 576 + 9886503251 Carmichael 1729 2465 2821 + 9456330241 Catalan 42 132 58786 + 6564120420 compositorial 24 192 1728 + 696729600 congruent 20 21 24 + 9999990 constructible 10 12 20 + 8589803520 cube 27 216 512 + 9910665792 Cullen 20971521 Cunningham 10 24 48 + 9998200080 Curzon 18 21 30 + 199999605 cyclic 133 209 247 + 9999881 D-number 21 63 111 + 6727539 d-powerful 24 63 132 + 9998252 de Polignac 2465 4311 6021 + 99996401 decagonal 10 27 126 + 9997050217 deceptive 481 1729 2821 + 9768280201 deficient 10 21 27 + 9999950 dig.balanced 10 12 21 + 199999936 double fact. 48 3840 46080 + 3715891200 droll 72 240 800 + 9767485440 Duffinian 21 27 36 + 9999881 eban 30 36 40 + 6066066060 economical 10 21 27 + 20000000 emirpimes 511 629 1011 + 99992171 enlightened 2176 2304 23328 + 2377970784 equidigital 10 21 27 + 19999709 eRAP 20 24 1104 + 9998143700 esthetic 10 12 21 + 9898987876 Eulerian 120 247 47840 + 3464764515 evil 10 12 18 + 999999926 factorial 24 120 720 + 6227020800 fibodiv 9744 13010 23418 + 9441619425 Fibonacci 21 144 2584 14930352 Friedman 126 153 216 + 995364 frugal 243 512 1215 + 999968571 gapful 100 108 110 + 10000000000 Gilda 110 152 220 + 9410385642 Giuga 30 happy 10 70 100 + 10000000 harmonic 140 270 1638 + 9866368512 heptagonal 18 81 112 + 9998149140 hex 1141 1387 3781 + 9936180151 hexagonal 45 120 153 + 9999171820 highly composite 12 24 36 + 6983776800 hoax 84 234 308 + 99996350 Hogben 21 111 133 + 9988303423 house 13959 15962 32103 + 9991438059 hungry 144 hyperperfect 21 2133 176661 + 8589869056 iban 10 12 20 + 777770 iccanobiF 1053 idoneal 10 12 18 + 1848 impolite 512 inconsummate 63 84 195 + 999852 insolite 111 11112 111111111 + 1211162112 interprime 12 18 21 + 99999965 Jacobsthal 21 171 1365 1398101 Jordan-Polya 12 24 36 + 9953280000 junction 111 117 204 + 99999966 Kaprekar 45 999 2223 + 8888888889 katadrome 10 20 21 + 9876543210 Lehmer 133 247 481 + 9996982621 Leyland 54 100 320 + 2179768320 lonely 120 1341 1344 + 47326801 Lucas 18 322 5778 lucky 21 63 111 + 9999825 Lynch-Bell 12 24 36 + 9867312 magic 111 870 2465 + 9885306184 magnanimous 12 20 21 + 62202281 metadrome 12 18 24 + 12345679 modest 111 133 209 + 1999826829 Moran 18 21 27 + 99999965 Motzkin 21 113634 310572 narcissistic 153 370 407 + 472335975 nialpdrome 10 20 21 + 10000000000 nonagonal 24 111 204 + 9997902714 nude 12 24 36 + 499999968 O'Halloran 12 20 36 + 660 oban 10 12 18 + 999 octagonal 21 40 133 + 9996757776 odious 21 42 50 + 1000000000 palindromic 111 171 222 + 8998558998 pancake 407 704 2212 + 9991254121 panconsummate 10 12 18 + 3097 pandigital 21 108 114 + 9876543210 partition 30 42 135 + 4835271870 pentagonal 12 70 117 + 9999553032 perfect 8589869056 pernicious 10 12 18 + 9999950 Perrin 10 12 90 + 5209407 persistent 1067283945 1067284395 1067284935 + 9876543210 plaindrome 12 18 24 + 8888888889 Poulet 645 1387 1729 + 9640119601 power 27 36 81 + 10000000000 powerful 27 36 72 + 10000000000 practical 12 18 20 + 10000000 prim.abundant 12 18 20 + 99967959 primeval 10279 12345679 1012345678 primorial 30 210 2310 + 223092870 pronic 12 20 30 + 9999900000 Proth 81 209 225 + 9999613953 pseudoperfect 12 18 20 + 1000000 repdigit 111 222 333 + 999999999 repfigit 1104 2208 2580 298320 repunit 21 40 63 + 9988303423 Rhonda 4752 5439 5832 + 9821117252 Ruth-Aaron 24 50 126 + 9999280186 Saint-Exupery 60 480 780 + 9999974400 Sastry 715 13224 647238399 + 1568473680 self 20 42 108 + 999999926 self-describing 10313316 10333116 12153331 + 8822888888 semiprime 10 21 111 + 99993563 sliding 20 70 110 + 9250000000 Smith 27 378 576 + 99999050 sphenic 30 42 70 + 99999965 square 36 81 100 + 10000000000 star 35113 80041 143221 + 9992002033 straight-line 111 135 210 + 9876543210 strobogrammatic 111 888 8118 + 9901691066 subfactorial 1854 1334961 super Niven 10 12 20 + 10000000000 super-d 81 190 247 + 9999881 superabundant 12 24 36 + 6983776800 tau 12 18 24 + 1000000000 taxicab 1729 4104 32832 + 9978655232 tetrahedral 10 20 84 + 9993352005 tetranacci 108 147312 triangular 10 21 36 + 9999171820 tribonacci 24 81 504 + 53798080 trimorphic 24 375 624 + 999999999 uban 10 12 18 + 10000000000 Ulam 18 36 48 + 10000236 undulating 171 252 414 + 9090909090 unprimeable 200 204 320 + 10000000 untouchable 120 162 210 + 999990 upside-down 555 53575 58525 + 599951115 vampire 1260 1530 102510 + 9953948970 wasteful 12 18 20 + 9999990 weird 70 10792 11410 + 994490 Wieferich 22953 298389 2685501 + 3626459235 Woodall 63 80 4607 + 3486784400 Zeisel 1729 5069629 185245273 + 1017436249 Zuckerman 12 24 36 + 9991111212 Zumkeller 12 20 24 + 100000 zygodrome 111 222 333 + 9999999666