evil numbers

A number

is called evil if the sum of its binary digits is even and odious if the sum of its binary digits is odd.

For example, $23=(10111)_2$ is evil since the sum of its binary digits is 4.

For 2$">, the number of evil and odious numbers of with $d$ binary digits is the same, and the two sets have also the same sum, i.e., $3\cdot 2^{2d-4}-2^{d-3}$ .

The first evil numbers are 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60 more terms

The smallest 3 × 3 magic square whose entries are evil numbers is

18	45	9
15	24	33
39	3	30

Below, the spiral pattern of evil numbers up to 2500. See the page on prime numbers for an explanation and links to similar pictures.

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

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

Useful links Mathworld, Evil Number
OEIS, Sequence A001969

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

a-pointer 101 293 811 + 999996407 ABA 18 24 72 + 999670898 aban 10 12 15 + 1000000001 abundant 12 18 20 + 50000000 Achilles 72 108 288 + 999581472 admirable 12 20 24 + 99999844 alt.fact. 101 619 35899 + 442386619 alternating 10 12 18 + 989898989 amenable 12 17 20 + 999999997 amicable 284 1210 2620 + 999258676 anti-perfect 141635817 apocalyptic 192 222 226 + 29999 arithmetic 15 17 20 + 9999999 astonishing 15 27 204 + 235629162 automorphic 9376 7109376 12890625 + 212890625 balanced p. 53 257 263 + 999998789 Bell 15 4140 21147 + 190899322 bemirp 1061 1091 1601 + 199800091 betrothed 48 75 195 + 953056175 binomial 10 15 20 + 999916840 brilliant 10 15 169 + 999999859 c.decagonal 101 281 1711 + 999910111 c.heptagonal 43 71 106 + 999930772 c.nonagonal 10 136 190 + 999916840 c.octagonal 169 225 441 + 999255321 c.pentagonal 51 106 141 + 999850006 c.square 85 113 221 + 999983921 c.triangular 10 46 85 + 999892414 cake 15 130 232 + 999802434 Canada 125 581 8549 Carmichael 561 1105 8911 + 993905641 Catalan 132 429 1430 + 477638700 Chen 17 23 29 + 99999971 compositorial 24 192 1728 + 696729600 congruent 15 20 23 + 9999999 constructible 10 12 15 + 858993459 cube 27 125 216 + 997002999 Cullen 65 897 2049 + 838860801 Cunningham 10 15 17 + 1000000001 Curzon 18 29 30 + 199999965 cyclic 15 17 23 + 9999989 D-number 15 33 39 + 7043133 d-powerful 24 43 63 + 9999827 de Polignac 149 337 373 + 99999937 decagonal 10 27 85 + 999650497 deceptive 703 4187 6533 + 999888709 deficient 10 15 17 + 9999999 dig.balanced 10 12 15 + 199999936 double fact. 15 48 105 + 185794560 droll 72 240 4224 + 995328000 Duffinian 27 36 39 + 9999993 eban 30 34 36 + 66066066 economical 10 15 17 + 20000000 emirp 17 71 113 + 199999931 emirpimes 15 39 51 + 99999997 enlightened 250 2176 2304 + 761743661 equidigital 10 15 17 + 19999985 eRAP 20 24 170 + 999621955 esthetic 10 12 23 + 989898989 Eulerian 57 66 120 + 423281535 factorial 24 120 720 + 479001600 fibodiv 75 149 183 + 974999994 Fibonacci 34 89 144 + 701408733 Friedman 125 126 153 + 999163 frugal 125 243 343 + 999999189 gapful 105 108 120 + 1000000001 Gilda 29 78 330 + 786119998 Giuga 30 858 66198 good prime 17 29 53 + 199968443 happy 10 23 68 + 10000000 harmonic 270 1638 8190 + 714954240 Harshad 10 12 18 + 999999926 heptagonal 18 34 189 + 999870004 hex 169 469 547 + 999789841 hexagonal 15 45 66 + 999916840 highly composite 12 24 36 + 735134400 hoax 58 85 136 + 99999706 Hogben 43 57 111 + 999792781 Honaker 263 1049 1091 + 999831103 house 78 652 933 + 997739002 hungry 17 144 2003 + 33662541 hyperperfect 325 697 1333 + 941755189 iban 10 12 17 + 777777 iccanobiF 39 514 836 + 641840787 idoneal 10 12 15 + 1848 inconsummate 63 65 75 + 999993 insolite 111 1122112 111111111 interprime 12 15 18 + 99999998 Jacobsthal 43 85 683 + 715827883 Jordan-Polya 12 24 36 + 995328000 junction 101 105 111 + 99999968 Kaprekar 45 99 297 + 909090909 katadrome 10 20 30 + 987654320 Kynea 23 287 4223 + 268468223 Lehmer 15 51 85 + 999962185 Leyland 17 54 57 + 774840978 lonely 23 53 120 + 163710121 Lucas 18 29 123 + 599074578 lucky 15 33 43 + 9999963 Lynch-Bell 12 15 24 + 9867312 m-pointer 23 11621 12113 + 411221311 magic 15 34 65 + 997809119 magnanimous 12 20 23 + 882428665 metadrome 12 15 17 + 123456789 modest 23 29 39 + 999989991 Moran 18 27 45 + 99999699 Motzkin 51 323 2188 + 142547559 narcissistic 153 371 407 + 146511208 nialpdrome 10 20 30 + 999999998 nonagonal 24 46 75 + 999947674 nude 12 15 24 + 499999824 O'Halloran 12 20 36 + 924 oban 10 12 15 + 999 octagonal 40 65 96 + 999698565 Ormiston 1913 18397 19013 + 999999131 palindromic 33 66 77 + 1000000001 palprime 101 353 373 + 999727999 pancake 29 46 92 + 999961561 panconsummate 10 12 15 + 1093 pandigital 15 75 78 + 381367044 partition 15 30 77 + 952050665 pentagonal 12 51 92 + 999840777 pernicious 10 12 17 + 9437184 Perrin 10 12 17 + 822261415 Pierpont 17 163 257 + 725594113 plaindrome 12 15 17 + 889999999 Poulet 561 645 1105 + 999828727 power 27 36 125 + 999824400 powerful 27 36 72 + 999824400 practical 12 18 20 + 10000000 prim.abundant 12 18 20 + 99999844 prime 17 23 29 + 999999937 primeval 113 1013 1037 + 102345679 primorial 30 210 2310 + 223092870 pronic 12 20 30 + 999982506 Proth 17 33 57 + 999948289 pseudoperfect 12 18 20 + 999999 rare 65 281089082 repdigit 33 66 77 + 888888888 repfigit 75 197 742 + 129572008 repunit 15 40 43 + 999792781 Rhonda 2835 4752 5664 + 999445823 Ruth-Aaron 15 24 77 + 999857816 Saint-Exupery 60 480 780 + 994882500 Sastry 183 528 715 + 924910143 self 20 53 75 + 999999983 self-describing 4444 666666 10123133 + 88888888 semiprime 10 15 33 + 99999997 sliding 20 29 65 + 1000000001 Smith 27 58 85 + 99999920 Sophie Germain 23 29 53 + 999998801 sphenic 30 66 78 + 99999986 square 36 144 169 + 999824400 star 337 937 1093 + 999156721 straight-line 111 123 135 + 888888888 strobogrammatic 96 101 111 + 1000000001 strong prime 17 29 71 + 99999547 subfactorial 1854 176214841 super Niven 10 12 20 + 999099900 super-d 105 106 119 + 9999981 superabundant 12 24 36 + 735134400 tau 12 18 24 + 999999920 taxicab 4104 32832 39312 + 998951616 tcefrep 498906 20671542 tetrahedral 10 20 120 + 989928820 tetranacci 15 29 108 + 747044834 triangular 10 15 36 + 999916840 tribonacci 24 149 504 + 615693474 trimorphic 24 51 75 + 925781249 truncatable prime 17 23 29 + 999636997 twin 17 29 43 + 999999193 uban 10 12 15 + 1000000001 Ulam 18 36 48 + 10000241 undulating 101 141 202 + 989898989 unprimeable 204 320 325 + 10000000 untouchable 96 120 210 + 999996 upside-down 46 159 258 + 999654111 vampire 1260 1530 1827 + 96977920 wasteful 12 18 20 + 9999999 weak prime 23 43 83 + 99999971 weakly prime 294001 604171 1062599 + 998839951 weird 836 4030 7912 + 997570 Wieferich 1093 3279 10533 + 725291847 Woodall 17 23 63 + 872415231 Zeisel 105 1419 1885 + 994732211 Zuckerman 12 15 24 + 997411968 Zumkeller 12 20 24 + 100000 zygodrome 33 66 77 + 999999988