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odious numbers
A number  $n$  is called odious if the sum of its binary digits is odd and evil if the sum of its binary digits is even.

For example,  $21=(10101)_2$  is odious since the sum of its binary digits is 3.

For  $d>2$, 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.,  $3\cdot 2^{2d-4}-2^{d-3}$.

A few numbers which are equal to the sum of their odious proper divisors are known. They are 28, 496, 8128, 415800, 2096128, 33550336, and 8589869056. They could be called odious-perfect numbers and it is worth to note that two of them, 415800 and 2096128, are not perfect numbers.

The first odious numbers are 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55 more terms

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

spiral pattern of odious numbers

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

a-pointer 11 13 103 + 999980231 ABA 32 50 64 + 999939200 aban 11 13 14 + 1000000000 abundant 42 56 70 + 49999998 Achilles 200 392 648 + 999939200 admirable 42 56 70 + 99999956 alt.fact. 19 4421 326981 36614981 alternating 14 16 21 + 989898985 amenable 13 16 21 + 1000000000 amicable 220 1184 2924 + 999728396 anti-perfect 244 285 133857 apocalyptic 157 218 220 + 30000 arithmetic 11 13 14 + 9999998 astonishing 1353 1863 3388 + 793039833 automorphic 25 76 376 + 787109376 balanced p. 157 173 211 + 999996329 Bell 52 203 877 bemirp 10091 16001 19001 + 199680881 betrothed 140 1648 2024 + 991087064 binomial 21 28 35 + 999961560 brilliant 14 21 25 + 999999863 c.decagonal 11 31 61 + 999627301 c.heptagonal 22 148 253 + 999575851 c.nonagonal 28 55 91 + 999782686 c.octagonal 25 49 81 + 999887641 c.pentagonal 16 31 76 + 999950001 c.square 13 25 41 + 999894481 c.triangular 19 31 64 + 999969871 cake 26 42 64 + 998152597 Canada 16999 Carmichael 1729 2465 2821 + 993420289 Carol 47 223 959 + 268402687 Catalan 14 42 4862 + 129644790 Chen 11 13 19 + 99999839 compositorial 207360 congruent 13 14 21 + 9999998 constructible 16 32 64 + 536870912 cube 64 512 2197 + 1000000000 Cullen 25 161 385 + 402653185 Cunningham 26 28 31 + 999999999 Curzon 14 21 26 + 199999974 cyclic 11 13 19 + 9999997 D-number 21 69 87 + 7042881 d-powerful 224 262 283 + 9999865 de Polignac 127 251 331 + 99999897 decagonal 52 370 451 + 999903451 deceptive 91 259 451 + 999670027 deficient 11 13 14 + 9999998 dig.balanced 11 19 21 + 199999932 double fact. 10395 654729075 droll 672 800 2240 + 930349056 Duffinian 16 21 25 + 9999997 eban 32 42 44 + 66066064 economical 11 13 14 + 19999999 emirp 13 31 37 + 199999061 emirpimes 26 49 62 + 99999982 enlightened 256 2048 2500 + 358722675 equidigital 11 13 14 + 19999999 eRAP 98 1104 1274 + 999751103 esthetic 21 32 56 + 989898789 Eulerian 11 26 247 + 848090912 factorial 3628800 39916800 fibodiv 14 19 28 + 986999390 Fibonacci 13 21 55 + 165580141 Friedman 25 121 127 + 999964 frugal 128 256 512 + 999994891 gapful 100 110 121 + 1000000000 Gilda 49 110 152 + 271953103 Giuga 1722 good prime 11 37 41 + 199968539 happy 13 19 28 + 9999992 harmonic 28 140 496 + 995248800 Harshad 21 42 50 + 999999999 heptagonal 55 81 112 + 999970000 hex 19 37 61 + 999899377 hexagonal 28 91 276 + 999827403 highly composite 83160 166320 332640 + 665280 hoax 22 84 94 + 99999332 Hogben 13 21 31 + 999982507 Honaker 131 457 1039 + 999843421 house 32 155 271 + 984609228 hungry 74 82810 7339199 hyperperfect 21 28 301 + 974380921 iban 11 14 21 + 777774 iccanobiF 13 124 1053 + 522619163 idoneal 13 16 21 + 385 impolite 16 32 64 + 536870912 inconsummate 62 84 161 + 999995 insolite 11112 122121216 interprime 21 26 42 + 99999965 Jacobsthal 11 21 171 + 357913941 Jordan-Polya 16 32 64 + 967458816 junction 103 107 109 + 99999970 Kaprekar 55 2223 2728 + 999999999 katadrome 21 31 32 + 987654321 Kynea 79 1087 16639 + 67125247 Lehmer 91 133 247 + 999903451 Leyland 32 100 145 + 387426321 lonely 211 1341 1342 + 325737821 Lucas 11 47 76 + 969323029 lucky 13 21 25 + 9999997 Lynch-Bell 124 128 162 + 9782136 m-pointer 61 1123 1231 + 311221111 magic 369 505 671 + 995433385 magnanimous 11 14 16 + 995955112 metadrome 13 14 16 + 23456789 modest 13 19 26 + 999999999 Moran 21 42 84 + 99999965 Motzkin 21 127 835 + 400763223 narcissistic 370 1634 548834 + 912985153 nialpdrome 11 21 22 + 1000000000 nonagonal 261 651 750 + 999829359 nude 11 22 44 + 499999968 O'Halloran 44 84 140 oban 11 13 16 + 998 octagonal 21 133 176 + 999917633 Ormiston 1931 18379 35671 + 999997697 palindromic 11 22 44 + 999999999 palprime 11 131 151 + 999565999 pancake 11 16 22 + 999916841 panconsummate 11 14 21 + 3097 pandigital 11 19 21 + 381366972 partition 11 22 42 + 679903203 pentagonal 22 35 70 + 999995690 perfect 28 496 8128 33550336 pernicious 11 13 14 + 9999998 Perrin 22 158 367 + 86700684 Pierpont 13 19 37 + 258280327 plaindrome 11 13 14 + 999999999 Poulet 341 1387 1729 + 998724481 power 16 25 32 + 1000000000 powerful 16 25 32 + 1000000000 practical 16 28 32 + 9999980 prim.abundant 42 56 70 + 99999956 prime 11 13 19 + 999999893 primeval 13 37 107 + 101234579 primorial 30030 510510 pronic 42 56 110 + 999792780 Proth 13 25 41 + 999981057 pseudoperfect 28 42 56 + 33550336 rare 621770 repdigit 11 22 44 + 999999999 repfigit 14 19 28 + 251133297 repunit 13 21 31 + 999982507 Rhonda 1568 5265 5439 + 984929951 Ruth-Aaron 16 25 49 + 999857817 Saint-Exupery 1620 7500 12960 + 997350120 Sastry 328 6099 13224 + 897506928 self 31 42 64 + 999999994 self-describing 22 224444 442244 + 66666622 semiprime 14 21 22 + 99999993 sliding 11 25 52 + 976664900 Smith 22 94 121 + 99999274 sphenic 42 70 110 + 99999994 square 16 25 49 + 999950884 star 13 37 73 + 999931141 straight-line 234 321 333 + 999999999 strobogrammatic 11 69 88 + 999689666 strong prime 11 37 41 + 99999959 subfactorial 44 265 14833 + 14684570 super Niven 50 70 100 + 1000000000 super-d 19 31 69 + 9999895 superabundant 166320 332640 665280 tau 56 84 88 + 1000000000 taxicab 1729 13832 20683 + 996762312 tcefrep 41673714 73687923 tetrahedral 35 56 84 + 999800616 tetranacci 56 208 5536 + 387559437 triangular 21 28 55 + 999961560 tribonacci 13 44 81 + 53798080 trimorphic 25 49 76 + 999999999 truncatable prime 13 31 37 + 999962683 twin 11 13 19 + 999999191 uban 11 13 16 + 1000000000 Ulam 11 13 16 + 10000175 undulating 121 131 151 + 929292929 unprimeable 200 206 208 + 9999988 untouchable 52 88 124 + 999968 upside-down 19 28 37 + 999951111 vampire 1395 1435 2187 + 96399072 wasteful 22 26 28 + 9999998 weak prime 13 19 31 + 99999941 weakly prime 505447 584141 971767 + 998740213 weird 70 5830 7192 + 999670 Wieferich 3511 7651 14209 + 748316985 Woodall 191 895 1214 + 838860799 Zeisel 1729 24211 54205 + 995762585 Zuckerman 11 112 115 + 997711344 Zumkeller 28 42 56 + 99988 zygodrome 11 22 44 + 999999999