nonagonal numbers

Nonagonal numbers are figurate numbers of the form $n(7n-5)/2$

(see picture aside).

If $n$ is a nonagonal number, then $7n+3$ is a triangular number.

The sum of the reciprocals of nonagonal numbers is quite involved:

$\begin{array}{c} \frac{4}{5}\left(\frac{\log{14}}{2}+\frac{\pi}{4}\tan\frac{3 \pi}{14}+\sin\left(\frac{\pi }{14}\right)\log \sin \frac{\pi}{7}+\cdots\hphantom{mmmm}\right.\\[4mm] \left.\hphantom{mmmm}\cdots+\cos\left(\frac{\pi }{7}\right)\log\cos\frac{3\pi }{14}+\sin\left(\frac{3 \pi }{14}\right)\log \sec \frac{\pi}{14}\right) \end{array}$

The first nonagonal numbers are 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089 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 nonagonal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Nonagonal Number
Wikipedia, Nonagonal number
OEIS, Sequence A001106

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

ABA 24 34171875 aban 24 46 75 + 999028000434 abundant 24 204 396 + 49999950 Achilles 28125 34171875 674528548509 admirable 24 474 alternating 325 474 856 + 987092541 amenable 24 204 261 + 999474456 apocalyptic 1639 2125 2301 + 29394 arithmetic 46 111 154 + 9992125 astonishing 204 betrothed 75 11145066075 binomial 325 969 82621 + 1353857339341 brilliant 559 41311 85879 + 979466221 c.decagonal 113740651 85504618351 c.nonagonal 325 82621 20985481 + 343874433963061 c.octagonal 1089 8281 978121 + 708304623404049 c.pentagonal 856 20026 17259391 + 8140708519516 c.square 3961 58825 255538225 + 244832844124441 c.triangular 46 228736 12750211 + 33478154327026 cake 1794 canyon 204 325 856 + 98710246 compositorial 24 congruent 24 46 111 + 9992125 constructible 24 204 Cunningham 24 325 22200 + 211162109090725 Curzon 261 1089 2301 + 197434146 cyclic 559 1639 3729 + 9944871 D-number 111 d-powerful 24 4959 7944 + 8978409 de Polignac 99541 125875 478225 + 96271975 decagonal 1212751 977965238701 deceptive 5045401 7756201 13072879 + 95329827199 deficient 46 75 111 + 9992125 dig.balanced 75 154 204 + 199648449 Duffinian 75 111 325 + 9885961 eban 46 34056 46000 4042000 economical 111 1216 1491 + 19769509 emirpimes 559 7291 11629 + 95831929 equidigital 111 1216 1491 + 19769509 eRAP 24 5811866875 esthetic 123234 evil 24 46 75 + 999947674 factorial 24 fibodiv 75 969 Friedman 139500 291601 293625 978121 frugal 28125 218125 1365625 + 988621354 gapful 154 396 1216 + 99989105139 happy 2484 4959 5781 + 9980301 Harshad 24 111 204 + 9997902714 heptagonal 26884 542041975 10928650279834 hex 12871 486019 5847005269 220785839821 hexagonal 325 5330229625 1353857339341 highly composite 24 hoax 6364 16975 17964 + 95027796 Hogben 111 651 99541 + 423236211589731 house 319118031 hyperperfect 325 iban 24 111 204 + 720271 idoneal 24 inconsummate 75 261 750 + 985536 insolite 111 interprime 111 154 969 + 99866019 Jordan-Polya 24 junction 111 204 14625 + 99679149 Kaprekar 71428071429 katadrome 75 651 750 Lehmer 69231 211191 550639 + 997375675591 lucky 75 111 261 + 9980301 Lynch-Bell 24 396 magic 111 metadrome 24 46 modest 46 111 4699 + 1722554325 Moran 111 261 7944 mountain 154 261 396 + 124895431 nialpdrome 75 111 651 + 88877542221 nude 24 111 396 + 499946184 O'Halloran 204 oban 75 325 396 + 969 octagonal 631125 286703855361 odious 261 651 750 + 999829359 palindromic 111 474 969 + 699030030996 pancake 46 154 11629 + 162573185113021 panconsummate 24 pandigital 75 170391 598851 + 9762483051 pentagonal 651 180868051 95317119801 pernicious 24 261 651 + 9956674 persistent 3906714825 6754039821 12053194786 + 98243361750 plaindrome 24 46 111 + 223399 Poulet 129889 2288661 4650049 + 999971076292381 power 1089 8281 978121 + 5996832038649 powerful 1089 8281 28125 + 708304623404049 practical 24 204 396 + 9909504 prim.abundant 474 254475 1279575 + 66675375 Proth 1089 74241 4788225 + 19633799169 pseudoperfect 24 204 396 + 985536 repdigit 111 6666 repfigit 75 repunit 111 651 99541 + 423236211589731 Rhonda 2455212471 172122239521 Ruth-Aaron 24 154 8281 + 866350730544 Saint-Exupery 4200 self 75 154 1491 + 999237889 self-describing 1041184144 12351833511751 semiprime 46 111 559 + 99268651 Smith 2484 4959 78375 + 95685471 sphenic 154 474 651 + 99716509 square 1089 8281 978121 + 708304623404049 star 215604181 straight-line 111 6666 strobogrammatic 111 super Niven 24 204 4200 + 30060063000 super-d 261 969 5226 + 9862446 superabundant 24 tau 24 204 396 + 991564704 tetrahedral 969 234136 triangular 325 82621 20985481 + 343874433963061 tribonacci 24 trimorphic 24 75 uban 46 75 5040039000075 + 56000046000009 Ulam 2484 2674 9699 + 9968484 undulating 474 969 595959 unprimeable 204 325 1350 + 9992125 untouchable 474 750 1956 + 985536 upside-down 46 158259 171939 + 698214 vampire 2189713059 wasteful 24 46 75 + 9992125 Zuckerman 24 111 Zumkeller 24 204 396 + 98364 zygodrome 111 5500 6666 + 990011222550099