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Curzon numbers
J.J.Tattersall defined the Curzon numbers to be those  $n$  for which  $2n+1$  divides  $2^n+1$.

The definition can be generalized, with respect to an arbitrary base  $k$, as those numbers for which  $k\cdot n+1$  divides  $k^n+1$.

Up to  $10^8$  there are 5540146 Curzon numbers.

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

11334165
125173221
2815233

Below, the spiral pattern of Curzon numbers up to  $100^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of Curzon numbers

The first Curzon numbers are 1, 2, 5, 6, 9, 14, 18, 21, 26, 29, 30, 33, 41, 50, 53, 54, 65, 69, 74, 78, 81, 86, 89, 90, 98, 105 more terms

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

a-pointer 293 1409 5333 + 199816073 ABA 18 50 81 + 199880018 aban 14 18 21 + 199000998 abundant 18 30 54 + 49999794 Achilles 1125 6125 36125 + 197506125 admirable 30 54 78 + 99999654 alt.fact. 326981 36614981 alternating 14 18 21 + 189898965 amenable 21 29 33 + 199999973 amicable 12285 87633 100485 + 188953970 anti-perfect 285 apocalyptic 245 278 285 + 29990 arithmetic 14 21 29 + 9999990 astonishing 429 1353 3882813 124015749 balanced p. 53 173 593 + 199999661 bemirp 1601 1901 10061 + 199680881 betrothed 1925 1279950 2576945 + 175742294 binomial 21 78 105 + 199970001 brilliant 14 21 209 + 199998101 c.decagonal 281 1901 4061 + 189758801 c.heptagonal 386 638 953 + 193035158 c.octagonal 81 441 729 + 198218241 c.pentagonal 141 1626 6126 + 198269826 c.square 41 113 221 + 199540265 cake 26 378 470 + 195151790 Canada 125 581 8549 Carmichael 561 2465 151530401 174352641 Catalan 14 429 1430 + 129644790 Chen 29 41 53 + 99996821 congruent 14 21 29 + 9999990 constructible 30 510 8738 + 6684774 cube 125 729 9261 + 161878625 Cullen 65 2049 10241 2228225 Cunningham 26 33 50 + 199883045 cyclic 29 33 41 + 9999869 D-number 21 33 69 + 7042569 d-powerful 89 153 209 + 9994326 de Polignac 509 809 905 + 99999041 deceptive 100001 1163801 1398101 + 186553961 deficient 14 21 26 + 9999981 dig.balanced 21 41 50 + 199999749 double fact. 105 34459425 Duffinian 21 50 65 + 9999921 eban 30 50 54 + 66064046 economical 14 21 29 + 19999953 emirp 113 761 953 + 199999661 emirpimes 26 158 221 + 99999794 equidigital 14 21 29 + 19999953 eRAP 98 1274 4233 + 192360321 esthetic 21 54 65 + 123434565 Eulerian 26 1013 131054 + 198410786 evil 18 29 30 + 199999965 fibodiv 14 366 13010 + 145624910 Fibonacci 21 89 233 + 196418 Friedman 125 153 1530 + 995346 frugal 125 729 1458 + 199998125 gapful 105 165 198 + 199999965 Gilda 29 78 330 + 977909 Giuga 30 good prime 29 41 53 + 199883693 happy 86 230 293 + 9999954 harmonic 270 1638 8190 + 81695250 Harshad 18 21 30 + 199999605 heptagonal 18 81 189 + 199393506 hexagonal 153 378 561 + 199590210 hoax 438 645 650 + 99996146 Hogben 21 273 813 + 199953741 Honaker 1049 2141 2273 + 199820213 house 78 933 20525 + 197241353 hungry 74 hyperperfect 21 19521 iban 14 21 41 + 777714 iccanobiF 9679838 idoneal 18 21 30 + 1365 inconsummate 65 173 261 + 999978 interprime 18 21 26 + 99999965 Jacobsthal 21 341 1365 + 89478485 junction 105 113 210 + 99999869 Kaprekar 4950 77778 95121 + 961038 katadrome 21 30 41 + 98765421 Lehmer 561 2465 8481 + 184504881 Leyland 54 593 1649 + 177264449 lonely 53 1341 24281 + 2010806 Lucas 18 29 3010349 54018521 lucky 21 33 69 + 9998841 Lynch-Bell 1926 9126 37926 m-pointer 1111361 9111341 magic 65 369 870 + 193710609 magnanimous 14 21 29 + 62202281 metadrome 14 18 26 + 12345678 modest 26 29 69 + 199804878 Moran 18 21 114 + 99999965 Motzkin 21 5798 narcissistic 153 548834 nialpdrome 21 30 33 + 99999981 nonagonal 261 1089 2301 + 197434146 nude 33 366 1626 + 199996326 oban 18 26 29 + 998 octagonal 21 65 341 + 199691525 odious 14 21 26 + 199999974 Ormiston 85313 93113 139109 + 199977773 palindromic 33 141 393 + 189959981 palprime 12821 14741 96269 + 187939781 pancake 29 254 326 + 197299181 panconsummate 14 18 21 + 413 pandigital 21 78 114 + 16434369 partition 30 1958 8349 + 2679689 pentagonal 210 330 590 + 199578570 pernicious 14 18 21 + 9999705 Perrin 29 90 158 + 57918 plaindrome 14 18 26 + 177777789 Poulet 341 561 645 + 181285001 power 81 125 441 + 198218241 powerful 81 125 441 + 198218241 practical 18 30 54 + 9999990 prim.abundant 18 30 78 + 99999654 prime 29 41 53 + 199999853 primeval 113 1013 10012349 primorial 30 210 2310 pronic 30 90 210 + 198965130 Proth 33 41 65 + 199720961 pseudoperfect 18 30 54 + 999978 rare 65 repdigit 33 333333 33333333 repfigit 14 7385 93993 repunit 21 273 341 + 199953741 Rhonda 5265 8526 15698 + 181585326 Ruth-Aaron 50 78 105 + 199894890 self 53 86 165 + 199999749 self-describing 10183133 10311533 10313318 + 42252425 semiprime 14 21 26 + 99999794 sliding 29 65 254 + 97666490 Smith 378 438 645 + 99998150 sphenic 30 78 105 + 99999965 square 81 441 729 + 198218241 straight-line 210 321 345 + 33333333 strobogrammatic 69 818 986 + 199609661 strong prime 29 41 281 + 99996821 subfactorial 1854 super Niven 30 50 90 + 101000010 super-d 69 81 105 + 9999981 tau 18 441 1089 + 195692121 taxicab 216125 3587409 38253878 + 185840109 tcefrep 498906 tetrahedral 165 2925 3654 + 192937325 tetranacci 29 triangular 21 78 105 + 199970001 tribonacci 81 trimorphic 125 249 749 + 25781249 truncatable prime 29 53 113 + 169956113 twin 29 41 281 + 199982969 uban 18 21 26 + 99000090 Ulam 18 26 53 + 10000214 undulating 141 393 414 + 161616161 unprimeable 326 510 530 + 9999954 untouchable 146 210 306 + 999830 upside-down 1289 1469 1649 + 982821 vampire 1530 105210 105750 + 95438970 wasteful 18 26 30 + 9999990 weak prime 89 113 233 + 99999353 weakly prime 3326489 7469789 14075273 + 162123869 weird 13790 16310 16730 + 977690 Wieferich 10533 52665 127881 + 6161805 Woodall 4373 590489 Zeisel 105 4505 59081 + 92458529 Zuckerman 13113 119313 173313 + 197111313 Zumkeller 30 54 78 + 99966 zygodrome 33 1133 1166 + 119999333