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lucky numbers
Lucky numbers are those numbers which survive a sieving process which is similar to the Eratosthenes sieve that can be used to compute the prime numbers.

Let us declare that 1 is a lucky number and let us start with a sieve containing only the odd numbers:

\[
1,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13,\ 15,\ 17,\ 19,\ 21,\ 23,\ 25,\dots
\]
The first number greater than 1 is 3, so we declare 3 lucky and we delete from the sieve all the numbers in a position which is a multiple of 3, i.e., 5, 11, 17, 23, and so on. We are left with
\[
1,\ 3,\ 7,\ 9,\ 13,\ 15,\ 19,\ 21,\ 25,\dots
\]
The first new survivor is 7, so we declare 7 lucky and we delete from the sieve all the numbers in a position which is a multiple of 7, i.e., 19, 39, and so on. The next lucky number is thus 9 and continuing this ideal process we could identify all the lucky numbers.

The lucky numbers have been studied because they have some properties (like density) which resemble the properties of primes.

The first lucky numbers are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105 more terms

Below, the spiral pattern of lucky numbers up to 3600. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of lucky numbers

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

a-pointer 13 631 1801 + 9923227 ABA 1029 2187 10125 + 7381125 aban 13 15 21 + 9000985 abundant 1575 2835 3465 + 9999825 Achilles 1323 7803 8575 + 9948123 admirable 4095 6435 9555 + 6220665 alt.fact. 619 alternating 21 25 43 + 9898927 amenable 13 21 25 + 9999997 amicable 12285 14595 69615 + 4482765 anti-perfect 285 133857 apocalyptic 285 361 529 + 29995 arithmetic 13 15 21 + 9999997 astonishing 15 429 133533 1301613 automorphic 25 90625 balanced p. 211 1123 3307 + 9983977 Bell 15 21147 bemirp 1606081 betrothed 75 195 1575 + 3676491 binomial 15 21 105 + 9988215 brilliant 15 21 25 + 9999811 c.decagonal 31 151 211 + 9779011 c.heptagonal 43 463 841 + 9731947 c.nonagonal 1711 3403 3655 + 9916831 c.octagonal 25 49 169 + 9941409 c.pentagonal 31 51 141 + 9726891 c.square 13 25 421 + 9994921 c.triangular 31 235 361 + 9957529 cake 15 93 23479 + 9437505 Carmichael 1105 2821 6601 + 9890881 Carol 223 Catalan 429 Chen 13 31 37 + 9999397 congruent 13 15 21 + 9999997 constructible 15 51 1285 983055 cube 729 29791 50653 + 9393931 Cullen 25 385 897 4609 Cunningham 15 31 33 + 9884737 Curzon 21 33 69 + 9998841 cyclic 13 15 31 + 9999997 D-number 15 21 33 + 7043109 d-powerful 43 63 135 + 9996789 de Polignac 127 331 997 + 9999997 decagonal 297 451 855 + 9942985 deceptive 259 451 2821 + 9890881 deficient 13 15 21 + 9999997 dig.balanced 15 21 37 + 9999963 double fact. 15 105 135135 Duffinian 21 25 49 + 9999997 economical 13 15 21 + 9999823 emirp 13 31 37 + 9999739 emirpimes 15 49 51 + 9999721 enlightened 119911 3720087 equidigital 13 15 21 + 9999823 eRAP 2079 14905 35167 + 9472585 esthetic 21 43 67 + 9876787 Eulerian 152637 evil 15 33 43 + 9999963 fibodiv 75 427 6505 + 3769767 Fibonacci 13 21 1597 + 75025 Friedman 25 127 289 + 995347 frugal 729 1029 1369 + 9985759 gapful 105 135 195 + 9997135 Gilda 49 997 good prime 37 67 127 + 9993223 happy 13 31 49 + 9999553 Harshad 21 63 111 + 9999825 heptagonal 189 235 2059 + 9967027 hex 37 127 169 + 9975457 hexagonal 15 231 2415 + 9988215 hoax 319 361 391 + 9998311 Hogben 13 21 31 + 9976123 Honaker 1039 1933 2221 + 9985231 house 933 1285 23101 + 9566817 hyperperfect 21 2133 19521 + 3420301 iban 21 43 73 + 777777 iccanobiF 13 1053 104937 idoneal 13 15 21 + 1365 inconsummate 63 75 195 + 999951 insolite 111 interprime 15 21 69 + 9999957 Jacobsthal 21 43 171 + 699051 junction 105 111 115 + 9999961 Kaprekar 99 297 9999 + 9372385 katadrome 21 31 43 + 9876541 Kynea 79 1087 16639 Lehmer 15 51 133 + 9973693 Leyland 131361 268705 lonely 211 31429 203713 + 2010807 Lucas 3571 9349 710647 Lynch-Bell 15 135 735 + 9315 m-pointer 1123 1231 15121 4121113 magic 15 111 1105 + 9951391 magnanimous 21 25 43 + 6640063 metadrome 13 15 25 + 2345689 modest 13 49 69 + 9992727 Moran 21 63 111 + 9999283 Motzkin 21 51 127 + 2356779 narcissistic 92727 nialpdrome 21 31 33 + 9999997 nonagonal 75 111 261 + 9980301 nude 15 33 99 + 9991395 oban 13 15 25 + 997 octagonal 21 133 645 + 9857281 odious 13 21 25 + 9999997 Ormiston 18397 36997 37813 + 9953071 palindromic 33 99 111 + 9986899 palprime 151 727 787 + 9514159 pancake 37 67 79 + 9965881 panconsummate 15 21 31 + 3097 pandigital 15 21 75 + 9998163 partition 15 135 231 + 5392783 pentagonal 51 477 651 + 9867555 pernicious 13 21 25 + 9999997 Perrin 51 367 1497 + 727653 Pierpont 13 37 73 + 5038849 plaindrome 13 15 25 + 8899999 Poulet 645 1105 1387 + 9920401 power 25 49 169 + 9941409 powerful 25 49 169 + 9948123 prim.abundant 1575 3465 4095 + 9999825 prime 13 31 37 + 9999823 primeval 13 37 10279 12379 Proth 13 25 33 + 9990145 pseudoperfect 1575 2835 3465 + 999075 repdigit 33 99 111 + 7777777 repfigit 75 7909 355419 1084051 repunit 13 15 21 + 9976123 Rhonda 2835 19435 33475 + 9843255 Ruth-Aaron 15 25 49 + 9997417 Sastry 106755 self 31 75 211 + 9999807 semiprime 15 21 25 + 9999997 sliding 25 133 205 + 7073125 Smith 319 391 483 + 9998613 sphenic 105 195 231 + 9999961 square 25 49 169 + 9941409 star 13 37 73 + 9746101 straight-line 111 135 159 + 7777777 strobogrammatic 69 111 619 + 6911169 strong prime 37 67 79 + 9999823 super-d 31 69 105 + 9999807 tau 4761 12321 15129 + 9941409 taxicab 46683 149389 216027 + 7882245 tetrahedral 1771 4495 9139 + 9886435 tetranacci 15 10671 triangular 15 21 105 + 9988215 tribonacci 13 927 1705 10609 trimorphic 25 49 51 + 9218751 truncatable prime 13 31 37 + 9981373 twin 13 31 43 + 9999049 uban 13 15 21 + 9000079 Ulam 13 69 87 + 9999823 undulating 141 151 171 + 9696969 unprimeable 535 895 1645 + 9999855 upside-down 37 73 159 + 997311 vampire 1395 1435 1827 + 792585 wasteful 33 51 63 + 9999997 weak prime 13 31 43 + 9999049 weakly prime 604171 2474431 6463267 Wieferich 1093 10533 14209 + 3697083 Woodall 63 159 895 + 524287 Zeisel 105 1419 5719 + 8712985 Zuckerman 15 111 115 + 7133175 Zumkeller 1575 2835 3465 + 97335 zygodrome 33 99 111 + 9999555