Smith numbers

A Smith number is a composite numbers $n$

such that sum of digits of $n$

is equal to the sum of digits of the prime factors of $n$

, counted with multiplicity. They are also called joke numbers.

For example, 666 is a Smith number since $666=2\cdot3\cdot3\cdot37$ and $6+6+6=2+3+3+3+7$ .

Note that hoax numbers are defined similarly, but they do not take into account prime factors multiplicity. Clearly, a squarefree hoax number is also a Smith number.

The first pair, triple, etc. of consecutive Smith numbers (also called Smith brothers) starts at 728, 73615, 4463535, 15966114, 2050918644, 164736913905, and 8090674745553, respectively.

If the requisite of compositeness is left aside, then all the prime numbers trivially satisfy the condition on the sum of digits and it is possible to find a run of length 9 starting at 1979414080360.

Below, the spiral pattern of Smith numbers up to $80^2$ . See the page on prime numbers for an explanation and links to similar pictures.

The smallest 3 × 3 magic square made of Smith numbers is

202	526	94
166	274	382
454	22	346

The first Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438 more terms

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

A graph displaying how many Smith numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Smith Number
Wikipedia, Smith number
OEIS, Sequence A006753
OEIS, Sequence A059754 (Smith brothers)

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

ABA 648 2888 9522 + 99630728 aban 22 27 58 + 99000990 abundant 378 438 576 + 49999544 Achilles 648 2888 17496 + 99630728 admirable 438 654 762 + 99994614 alternating 27 58 85 + 98987250 amenable 85 121 265 + 99999920 amicable 66992 879712 1392368 + 91996816 anti-perfect 133857 apocalyptic 355 382 648 + 29960 arithmetic 22 27 85 + 9999895 astonishing 27 7119 betrothed 507759 544784 1057595 + 84854315 binomial 378 666 861 + 99906180 brilliant 121 319 391 + 97333987 c.decagonal 9031 30811 34861 + 98323951 c.heptagonal 22 3046 24403 + 99263326 c.nonagonal 29890 32896 141778 + 99680140 c.octagonal 121 729 10201 + 96530625 c.pentagonal 391 526 1626 + 98486131 c.square 85 265 2965 + 97482685 c.triangular 85 166 274 + 99621526 cake 378 576 215930 + 97031176 Carmichael 656601 27336673 37167361 93614521 Catalan 9694845 congruent 22 85 94 + 9999895 constructible 85 4369 32896 + 786444 cube 27 729 19683 + 62099136 Cunningham 483 728 1935 + 99960003 Curzon 378 438 645 + 99998150 cyclic 85 265 319 + 9999895 D-number 1935 24963 178923 + 2215983 d-powerful 378 1255 2227 + 9992762 de Polignac 3505 4855 8023 + 99993307 decagonal 27 85 13747 + 99865045 deceptive 8149 226273 642001 + 98269633 deficient 22 27 58 + 9999922 dig.balanced 166 202 438 + 95580940 droll 37879808 Duffinian 27 85 121 + 9999355 eban 4054 6036 6054 + 66036036 economical 27 121 166 + 19999858 emirpimes 58 85 94 + 99999238 equidigital 27 121 166 + 19999858 eRAP 2079 976096 1444960 + 96954909 esthetic 121 454 654 + 98765432 Eulerian 152637 evil 27 58 85 + 99999920 fibodiv 1822 6178 22117 + 86184702 Fibonacci 1346269 Friedman 121 1255 3864 + 983445 frugal 729 10201 17496 + 99984843 gapful 121 1111 1376 + 99995670 happy 94 319 391 + 9999355 harmonic 2970 27846 Harshad 27 378 576 + 99999050 heptagonal 2839 10791 24354 + 99878121 hexagonal 378 861 2556 + 99906180 highly composite 2882880 hoax 22 58 85 + 99999706 Hogben 4557 44733 190533 + 99351057 house 25884 56560 258477 + 83980587 hyperperfect 4013833 8883841 14489437 + 79001833 iban 22 27 121 + 777417 idoneal 22 58 85 inconsummate 438 483 645 + 999536 interprime 274 483 636 + 99999058 Jacobsthal 85 Jordan-Polya 576 7077888 8957952 junction 202 319 517 + 99999160 Kaprekar 681318 851851 49995000 55636659 katadrome 85 94 654 + 98765432 Lehmer 85 1111 4369 + 99264451 Leyland 67137425 lonely 15704 47326803 lucky 319 391 483 + 9998613 Lynch-Bell 648 728 1935 + 9867312 magic 143781 578865 629910 + 96550565 magnanimous 58 85 94 + 66886483 metadrome 27 58 346 + 12456789 modest 627 654 666 + 99990243 Moran 27 645 666 + 99957093 nialpdrome 22 85 94 + 99999920 nonagonal 2484 4959 78375 + 95685471 nude 22 636 648 + 99993492 oban 27 58 85 + 985 octagonal 645 6816 9296 + 99867160 odious 22 94 121 + 99999274 palindromic 22 121 202 + 99144199 pancake 22 121 562 + 99426152 panconsummate 85 121 166 pandigital 2902 9015 19615 + 95580940 partition 22 627 1255 pentagonal 22 852 27270 + 99923285 pernicious 22 94 121 + 9999895 Perrin 22 plaindrome 22 27 58 + 66666668 Poulet 645 4369 13747 + 97863529 power 27 121 576 + 99281296 powerful 27 121 576 + 99630728 practical 378 576 588 + 9997840 prim.abundant 438 654 762 + 99994614 pronic 2970 8372 12656 + 99770132 Proth 1921 3649 7809 + 99581953 pseudoperfect 378 438 576 + 999950 repdigit 22 666 1111 6666666 repfigit 4788 86935 repunit 85 121 1111 + 99351057 Rhonda 47265 53742 133857 + 86148315 Ruth-Aaron 4185 4191 28449 + 99657775 Saint-Exupery 8696160 9982500 13618860 + 87567480 self 121 378 648 + 99999050 self-describing 22 15173331 17153133 + 42272724 semiprime 22 58 85 + 99999706 sliding 62660 sphenic 438 483 627 + 99997305 square 121 576 729 + 99281296 star 121 1633 16537 + 98925961 straight-line 654 666 852 + 98765432 strobogrammatic 1111 1881 168891 + 91688916 subfactorial 265 super-d 319 454 690 + 9999895 tau 636 852 1284 + 99999920 taxicab 402597 920673 1609272 + 95547816 tetrahedral 4960 9880 37820 + 95988256 triangular 378 666 861 + 99906180 tribonacci 274 trimorphic 7890625 uban 22 27 58 + 99000083 Ulam 319 382 483 + 9997341 undulating 121 202 454 + 90909090 unprimeable 535 895 1642 + 9999895 untouchable 562 576 852 + 999970 upside-down 654 852 5915 + 99682411 vampire 6880 117067 124483 + 94765918 wasteful 22 58 85 + 9999922 weird 73616 388076 512468 + 933812 Woodall 895 Zeisel 44544219 Zuckerman 1111 3168 7119 + 79817472 Zumkeller 378 438 588 + 99920 zygodrome 22 666 1111 + 99991155