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star numbers
Star numbers are figurate numbers (see picture aside) defined as  $S_n=6n(n-1)+1$. They coincide with centered dodecagonal numbers.

Three formulas involving star numbers:

\[\sum_{n=1}^{\infty}\frac{1}{S_n}= \frac{\pi\tan\left(\frac{\pi}{2\sqrt{3}}\right)}{2\sqrt{3}}\,,\quad \sum_{n=0}^{\infty}\frac{S_n}{n!}=7e\,, \quad\mbox{and}\quad\sum_{n=1}^{\infty}\frac{S_n}{2^n}=25\,. \]

The first star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093 more terms

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

A graph displaying how many star numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of star numbers
Useful links Wikipedia, Star number
Mathworld, Star Number
OEIS, Sequence A003154
Star numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 181 2053 + 9457731037 aban 13 37 73 + 987129000937 alternating 121 181 541 + 967054321 amenable 13 37 73 + 999931141 apocalyptic 541 937 1261 + 29821 arithmetic 13 37 73 + 9992341 balanced p. 3313 12973 15913 + 9942603337 binomial 253 49141 9533161 + 358770992581 brilliant 121 253 793 + 998847037 c.decagonal 661 318781 153651961 + 35696730969901 c.heptagonal 253 14113 3063061 + 448314542426521 c.nonagonal 253 49141 9533161 + 69599723176621 c.octagonal 121 11881 1164241 + 107341182792481 c.pentagonal 181 3901 695641 + 221235542824081 c.square 13 181 2521 + 692665874901013 Carmichael 258634741 Chen 13 37 181 + 98050837 congruent 13 37 181 + 9992341 cube 226981 Cunningham 37 3601 352837 + 32530984516837 cyclic 13 37 73 + 9992341 d-powerful 2773 23437 26533 + 9563437 de Polignac 337 23437 31537 + 99267337 decagonal 14701 141164101 1355457688201 deceptive 14701 35113 976873 + 790673521 deficient 13 37 73 + 9992341 dig.balanced 37 541 661 + 199861273 Duffinian 121 253 793 + 9992341 economical 13 37 73 + 19994701 emirp 13 37 73 + 199238437 emirpimes 4537 7141 11881 + 99951853 equidigital 13 37 73 + 19994701 esthetic 121 evil 337 937 1093 + 999156721 Fibonacci 13 Friedman 121 226981 231673 + 683437 frugal 11881 226981 1164241 + 381077581 gapful 121 253 1441 + 99989241337 good prime 37 541 937 + 114607621 happy 13 793 937 + 9868837 Harshad 35113 80041 143221 + 9992002033 heptagonal 5221 20060473 77072333437 296111885005873 hex 37 1261 42841 + 65838103892821 hexagonal 49141 1849384153 69599723176621 hoax 1633 16537 43861 + 98925961 Hogben 13 73 1261 + 64514252901853 Honaker 11353 15913 20533 + 981120937 hyperperfect 3901 iban 73 121 1441 + 720373 iccanobiF 13 idoneal 13 37 253 inconsummate 3037 16537 57037 + 683437 interprime 8437 48061 70633 + 99169741 junction 2521 5221 12421 + 99951853 katadrome 73 541 9841 Lehmer 1261 7141 14701 + 896360752261 lucky 13 37 73 + 9746101 magnanimous 661 2281 metadrome 13 37 modest 13 433 1633 + 1702018837 Moran 2986381 6875821 7359337 + 97711561 nialpdrome 73 433 541 + 777777661 nonagonal 215604181 oban 13 37 73 + 937 octagonal 1884961 2510229508033 odious 13 37 73 + 999931141 Ormiston 379513 465373 1491013 + 1992575713 palindromic 121 181 1441 + 396868131868693 palprime 181 12421 18481 + 396868131868693 pancake 37 121 7141 + 33193978321141 panconsummate 37 73 121 + 1093 pernicious 13 37 73 + 9992341 persistent 11456829037 12054876913 14082961537 + 94618251037 Pierpont 13 37 73 433 plaindrome 13 37 337 + 1122337 Poulet 49141 976873 2649361 + 175321803251881 power 121 11881 226981 + 1095432263641 powerful 121 11881 226981 + 107341182792481 prime 13 37 73 + 999991228573 primeval 13 37 Proth 13 repunit 13 73 121 + 308836698141973 Ruth-Aaron 3816037 3912337 19689193 + 47317435321 self 121 793 2773 + 990041221 semiprime 121 253 793 + 99951853 Smith 121 1633 16537 + 98925961 sphenic 8437 9361 15301 + 99756037 square 121 11881 1164241 + 107341182792481 strobogrammatic 181 strong prime 37 541 937 + 99364981 super-d 181 337 2281 + 9930493 triangular 253 49141 9533161 + 69599723176621 tribonacci 13 truncatable prime 13 37 73 + 924337 twin 13 73 181 + 972241021 uban 13 37 73 + 96000072000013 Ulam 13 253 11881 + 9639337 undulating 121 181 upside-down 37 73 wasteful 253 793 1441 + 9992341 weak prime 13 73 181 + 99853921 Wieferich 1093 zygodrome 220033 771133 332255533 77111779933