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pernicious numbers
A number  $n$  is called pernicious if it contains a prime number of ones in its binary representation.

For example,  $21=(10101)_2$  is pernicious since it containes 3 ones and 3 is a prime number.

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

7176
91011
14313

Below, the spiral pattern of pernicious numbers up to  $60^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of pernicious numbers

The first pernicious numbers are 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36 more terms

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

a-pointer 11 13 103 + 9999347 ABA 18 24 50 + 9999392 aban 10 11 12 + 9000992 abundant 12 18 20 + 9999980 Achilles 72 200 288 + 9999392 admirable 12 20 24 + 9999856 alt.fact. 19 4421 326981 alternating 10 12 14 + 9898985 amenable 12 13 17 + 9999997 amicable 220 1184 2924 + 9892936 anti-perfect 244 285 133857 apocalyptic 157 192 218 + 30000 arithmetic 11 13 14 + 9999998 astonishing 1353 1863 3388 + 7103835 automorphic 25 76 376 + 109376 balanced p. 157 173 211 + 9997219 Bell 52 203 877 bemirp 16001 19001 1109881 + 1889011 betrothed 48 140 1648 + 9247095 binomial 10 20 21 + 9939111 brilliant 10 14 21 + 9999811 c.decagonal 11 31 61 + 9989911 c.heptagonal 22 148 253 + 9978613 c.nonagonal 10 28 55 + 9890128 c.octagonal 25 49 81 + 9991921 c.pentagonal 31 76 181 + 9995001 c.square 13 25 41 + 9968113 c.triangular 10 19 31 + 9988471 cake 26 42 93 + 9886826 Canada 16999 Carmichael 1729 2465 2821 + 9890881 Carol 47 223 3967 + 1046527 Catalan 14 42 132 Chen 11 13 17 + 9999991 compositorial 24 192 207360 congruent 13 14 20 + 9999998 constructible 10 12 17 + 8912896 cube 2197 15625 17576 + 9663597 Cullen 25 65 161 + 4718593 Cunningham 10 17 24 + 9991922 Curzon 14 18 21 + 9999705 cyclic 11 13 17 + 9999997 D-number 21 33 69 + 7042881 d-powerful 24 132 224 + 9999865 de Polignac 127 251 331 + 9999997 decagonal 10 52 370 + 9993501 deceptive 91 259 451 + 9890881 deficient 10 11 13 + 9999998 dig.balanced 10 11 12 + 9999925 double fact. 48 384 10395 droll 72 672 800 + 8404992 Duffinian 21 25 35 + 9999997 eban 34 36 40 + 6066064 economical 10 11 13 + 9999991 emirp 13 17 31 + 9999971 emirpimes 26 49 62 + 9999998 enlightened 2176 2304 2500 + 2560000 equidigital 10 11 13 + 9999991 eRAP 20 24 98 + 9960047 esthetic 10 12 21 + 9898789 Eulerian 11 26 66 + 8388584 evil 10 12 17 + 9437184 factorial 24 3628800 fibodiv 14 19 28 + 9869390 Fibonacci 13 21 34 + 9227465 Friedman 25 121 127 + 997375 frugal 625 1029 1250 + 9999803 gapful 100 110 121 + 9999985 Gilda 49 110 152 + 3849154 Giuga 1722 good prime 11 17 37 + 9993337 happy 10 13 19 + 9999992 harmonic 28 140 496 + 8506400 Harshad 10 12 18 + 9999950 heptagonal 18 34 55 + 9967027 hex 19 37 61 + 9953587 hexagonal 28 66 91 + 9925740 highly composite 12 24 36 + 665280 hoax 22 84 94 + 9999895 Hogben 13 21 31 + 9988761 Honaker 131 457 1039 + 9985231 house 155 271 434 + 9867520 hungry 17 74 144 7339199 hyperperfect 21 28 301 + 9699181 iban 10 11 12 + 777774 iccanobiF 13 124 514 + 12815 idoneal 10 12 13 + 520 inconsummate 62 65 84 + 999995 insolite 11112 interprime 12 18 21 + 9999957 Jacobsthal 11 21 171 + 1398101 Jordan-Polya 12 24 36 + 9676800 junction 103 107 109 + 9999961 Kaprekar 55 2223 2728 + 5072059 katadrome 10 20 21 + 9876542 Kynea 79 1087 263167 4198399 Lehmer 91 133 247 + 9995189 Leyland 17 100 145 + 9865625 lonely 211 1341 1342 + 4652428 Lucas 11 18 47 + 7881196 lucky 13 21 25 + 9999997 Lynch-Bell 12 24 36 + 9782136 m-pointer 61 1123 1231 + 9111341 magic 34 65 260 + 9841635 magnanimous 11 12 14 + 9959374 metadrome 12 13 14 + 3456789 modest 13 19 26 + 9998991 Moran 18 21 42 + 9999950 Motzkin 21 127 835 + 2356779 narcissistic 370 1634 8208 + 9800817 nialpdrome 10 11 20 + 9999998 nonagonal 24 261 651 + 9956674 nude 11 12 22 + 9999288 O'Halloran 12 20 36 + 260 oban 10 11 12 + 998 octagonal 21 40 65 + 9955408 odious 11 13 14 + 9999998 Ormiston 1931 35879 35897 + 9990797 palindromic 11 22 33 + 9998999 palprime 11 131 151 + 9981899 pancake 11 22 37 + 9997157 panconsummate 10 11 12 + 3097 pandigital 11 19 21 + 800661 partition 11 22 42 + 9289091 pentagonal 12 22 35 + 9998795 perfect 28 496 8128 Perrin 10 12 17 + 9141872 Pierpont 13 17 19 + 8503057 plaindrome 11 12 13 + 8889999 Poulet 341 1387 1729 + 9995671 power 25 36 49 + 9998244 powerful 25 36 49 + 9999392 practical 12 18 20 + 9999980 prim.abundant 12 18 20 + 9999856 prime 11 13 17 + 9999991 primeval 13 37 107 + 1234579 primorial 510510 pronic 12 20 42 + 9976122 Proth 13 17 25 + 9994241 pseudoperfect 12 18 20 + 1000000 rare 65 621770 repdigit 11 22 33 + 6666666 repfigit 14 19 28 + 355419 repunit 13 21 31 + 9988761 Rhonda 1568 5265 5824 + 9541116 Ruth-Aaron 24 25 49 + 9978606 Saint-Exupery 1620 7500 12960 + 9903180 Sastry 328 528 13224 self 20 31 42 + 9999998 self-describing 22 224444 442244 444422 semiprime 10 14 21 + 9999998 sliding 11 20 25 + 9868025 Smith 22 94 121 + 9999895 sphenic 42 66 70 + 9999985 square 25 36 49 + 9998244 star 13 37 73 + 9992341 straight-line 234 258 321 + 7654321 strobogrammatic 11 69 88 + 9991666 strong prime 11 17 37 + 9999971 subfactorial 44 265 133496 1334961 super Niven 10 12 20 + 9903600 super-d 19 31 69 + 9999895 superabundant 12 24 36 + 665280 tau 12 18 24 + 9999992 taxicab 1729 4104 13832 + 8872487 tetrahedral 10 20 35 + 9810580 tetranacci 56 208 5536 + 2033628 triangular 10 21 28 + 9939111 tribonacci 13 24 44 + 1389537 trimorphic 24 25 49 + 7890625 truncatable prime 13 17 31 + 9986113 twin 11 13 17 + 9999973 uban 10 11 12 + 9000093 Ulam 11 13 18 + 9999977 undulating 121 131 151 + 9797979 unprimeable 200 206 208 + 9999988 untouchable 52 88 96 + 999968 upside-down 19 28 37 + 9995111 vampire 1395 1435 2187 + 815958 wasteful 12 18 20 + 9999998 weak prime 13 19 31 + 9999991 weakly prime 505447 584141 1524181 + 9016079 weird 70 5830 7192 + 999670 Wieferich 14209 42627 99463 + 6161805 Woodall 17 80 191 + 5764800 Zeisel 1729 54205 114985 + 8835799 Zuckerman 11 12 24 + 9813312 Zumkeller 12 20 24 + 99988 zygodrome 11 22 33 + 9999988