D-numbers

Also known as 3-Knödel numbers, they are number $n>3$

such that

divides $k^{n-2}-k$ for all $1<k<n$

relatively prime to $n$

For example, 9 is a D-number since it divides all the numbers $2^7-2$ , $4^7-4$ , $5^7-5$ , $7^7-7$ and $8^7-8$ .

The smallest 3 × 3 magic square whose entries are D-numbers is

87	267	69
123	141	159
213	15	195

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

The first D-numbers are 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201, 213, 219 more terms

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

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

Useful links Mathworld, D-Number
Wikipedia, Knödel number
OEIS, Sequence A033553

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

aban 15 21 33 39 + 7000989 abundant 4095 16695 1527435 2475795 + 6310395 admirable 4095 alternating 21 63 69 87 + 7038723 amenable 21 33 57 69 + 7043133 apocalyptic 411 669 693 723 + 29919 arithmetic 15 21 33 39 + 7043133 astonishing 15 Bell 15 betrothed 195 binomial 15 21 1953 3003 + 5572791 brilliant 15 21 c.pentagonal 51 141 681 951 + 7018251 cake 15 93 canyon 201 213 219 303 + 7012389 congruent 15 21 39 63 + 7043133 constructible 15 51 771 196611 Cullen 2049 Cunningham 15 33 63 129 + 7011903 Curzon 21 33 69 141 + 7042569 cyclic 15 33 51 69 + 7043109 d-powerful 63 267 849 2283 + 7040589 de Polignac 7431 8031 11541 17229 + 7042281 deficient 15 21 33 39 + 7043133 dig.balanced 15 21 141 177 + 7043079 double fact. 15 Duffinian 21 39 57 63 + 7043133 economical 15 21 111 123 + 2999949 emirpimes 15 39 51 93 + 7042731 equidigital 15 21 111 123 + 2999949 esthetic 21 87 123 321 + 5454321 Eulerian 57 1191 4083 evil 15 33 39 51 + 7043133 fibodiv 183 2733 3903 9267 + 4788039 Fibonacci 21 Friedman 15567 15627 16347 16743 + 995331 gapful 195 315 693 1443 + 6999363 happy 129 219 291 921 + 7042773 Harshad 21 63 111 195 + 6727539 heptagonal 21483 hexagonal 15 3003 hoax 24963 178923 226083 334971 + 3827043 Hogben 21 57 111 183 + 6993381 house 933 7617 hyperperfect 21 iban 21 111 123 141 + 777477 iccanobiF 39 idoneal 15 21 33 57 + 177 inconsummate 63 195 381 411 + 999993 insolite 111 interprime 15 21 39 69 + 7042323 Jacobsthal 21 junction 111 303 309 315 + 7042227 katadrome 21 51 63 87 + 987621 Lehmer 15 51 771 196611 Leyland 57 177 131361 423393 lonely 31431 370317 Lucas 123 843 271443 lucky 15 21 33 51 + 7043109 Lynch-Bell 15 315 1935 magic 15 111 magnanimous 21 metadrome 15 39 57 69 + 2346789 modest 39 69 111 309 + 7023333 Moran 21 63 111 195 + 2000001 Motzkin 21 51 853467 mountain 141 183 195 291 + 6987621 nialpdrome 21 33 51 63 + 6666531 nonagonal 111 nude 15 33 111 315 + 3313131 oban 15 33 39 57 + 993 octagonal 21 odious 21 69 87 93 + 7042881 palindromic 33 111 141 303 + 7023207 panconsummate 15 21 39 57 + 267 pandigital 15 21 141 177 + 799899 partition 15 pentagonal 51 pernicious 21 33 69 87 + 7042881 Perrin 39 51 1497 1983 + 2240877 plaindrome 15 33 39 57 + 6667779 prim.abundant 4095 16695 Proth 33 57 129 177 + 6987777 pseudoperfect 4095 16695 repdigit 33 111 repfigit 7647 repunit 15 21 57 63 + 6993381 Ruth-Aaron 15 1683 8463 1040403 Sastry 183 self 411 501 591 681 + 7042773 semiprime 15 21 33 39 + 7043133 Smith 1935 24963 178923 226083 + 2215983 sphenic 195 399 1023 1443 + 7011903 straight-line 111 123 159 321 + 876543 strobogrammatic 69 111 6009 68889 + 6911169 super-d 69 219 339 381 + 7042881 tetranacci 15 10671 283953 triangular 15 21 1953 3003 + 5572791 trimorphic 51 249 501 5001 + 500001 uban 15 21 33 39 + 7000041 Ulam 57 69 87 177 + 7042683 undulating 141 303 393 717 + 3737373 unprimeable 10815 188235 1527435 1700415 + 6340995 upside-down 159 753 951 112899 + 994611 wasteful 33 39 51 57 + 7043133 Wieferich 3279 10533 Woodall 63 159 1023 10239 Zuckerman 15 111 315 13113 Zumkeller 4095 16695 zygodrome 33 111 11199 22233 + 6666999