octagonal numbers

Octagonal numbers are figurate numbers of the form $n(3n-2)$

(see picture aside).

The sum of the reciprocals of octagonal numbers is $(\pi\sqrt{3}+9\log3)/12$ .

The first octagonal numbers are 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833 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 octagonal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Octagonal Number
Wikipedia, Octagonal number
OEIS, Sequence A000567

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

ABA 78408 752875208 7229107670408 aban 21 40 65 + 991898000133 abundant 40 96 176 + 49980008 Achilles 78408 128066400 752875208 + 7229107670408 admirable 40 11408 13736 alternating 21 65 96 + 947816325 amenable 21 40 65 + 999917633 apocalyptic 1160 1281 1541 + 29800 arithmetic 21 65 96 + 9999176 binomial 21 560 5985 + 1086210502741 brilliant 21 341 481 + 999260501 c.decagonal 4961 19061281 73233437761 281362848817601 c.heptagonal 736 5461 496133 + 151906034826008 c.nonagonal 1957283461 1086210502741 c.octagonal 225 43681 8473921 + 61866420601441 c.pentagonal 7701 219781 1789106881 + 415648888795141 c.square 481 8321 4617761 + 425750177334721 c.triangular 97921 1039585 130402572385 1384429704481 cake 176 1160 canyon 408 645 736 + 98762456 Carmichael 2465 2821 15841 + 952910081761 congruent 21 65 96 + 9977280 constructible 40 96 408 Cullen 65 Cunningham 65 1680 12545 + 461777249934008 Curzon 21 65 341 + 199691525 cyclic 65 133 341 + 9900833 D-number 21 d-powerful 2133 2465 4485 + 8923425 de Polignac 1541 2465 14981 + 97903681 decagonal 54405 2047494625 77055412679701 deceptive 481 2821 5461 + 99985191041 deficient 21 65 133 + 9988225 dig.balanced 21 225 936 + 199985345 Duffinian 21 65 133 + 9988225 eban 40 economical 21 133 1281 + 19979521 emirpimes 341 8321 109061 + 97219361 equidigital 21 133 1281 + 19979521 eRAP 7105 54153505 693363221 787190405 esthetic 21 65 evil 40 65 96 + 999698565 Fibonacci 21 Friedman 736 32865 87381 + 287680 frugal 22016 176661 196096 + 847392133 gapful 176 225 280 + 99999431280 happy 133 176 280 + 9727201 Harshad 21 40 133 + 9996757776 heptagonal 297045 69010153345 hexagonal 11781 113123361 1086210502741 highly composite 1680 hoax 645 2640 35425 + 98452865 Hogben 21 133 4033 + 36485471742721 hyperperfect 21 2133 19521 + 129127041 iban 21 40 341 + 373121 idoneal 21 40 133 + 408 inconsummate 65 645 936 + 990725 interprime 21 176 225 + 99728736 Jacobsthal 21 341 5461 + 375299968947541 Jordan-Polya 96 8640 junction 408 1825 5720 + 99971041 katadrome 21 40 65 + 8640 Lehmer 133 481 2465 + 999918240133 lucky 21 133 645 + 9857281 Lynch-Bell 936 3816 427896 magic 65 2465 magnanimous 21 65 736 2465 modest 133 833 1045 + 1976333333 Moran 21 133 481 645 Motzkin 21 mountain 176 280 341 + 467975320 nialpdrome 21 40 65 + 833333333333333 nonagonal 631125 286703855361 nude 936 3816 6816 + 499668696 oban 65 96 560 + 936 odious 21 133 176 + 999917633 palindromic 8008 120232021 124060421 + 12212500521221 pancake 1541 5461 14794081 + 503470501381 panconsummate 21 40 481 pandigital 21 225 12545 + 9053782416 partition 176 pentagonal 176 1575425 234631320 + 312461813932000 pernicious 21 40 65 + 9955408 persistent 14993207685 15539762408 17063925845 + 91528477360 plaindrome 133 225 13333 + 133333333333333 Poulet 341 645 2465 + 999993874626581 power 225 43681 8473921 + 318907548961 powerful 225 43681 78408 + 61866420601441 practical 40 96 176 + 9999176 prim.abundant 5985 13736 49665 + 66797445 Proth 65 225 481 + 824635817985 pseudoperfect 40 96 176 + 994176 rare 65 repunit 21 40 133 + 375299968947541 Ruth-Aaron 43681 1829627856 2900261761 + 970469500408 self 176 3008 3201 + 999808096 semiprime 21 65 133 + 98728033 sliding 65 133 Smith 645 6816 9296 + 99867160 sphenic 645 1045 1281 + 99971041 square 225 43681 8473921 + 61866420601441 star 1884961 2510229508033 strobogrammatic 96 8008 super Niven 40 280 408 + 606600 super-d 481 1281 4256 + 9977280 superabundant 1680 tau 40 96 225 + 998713056 tetrahedral 560 20337240 triangular 21 11781 203841 + 18793835590881 uban 21 40 65 + 78030051000008 Ulam 341 3605 5208 + 9792133 unprimeable 1680 3816 5720 + 9999176 untouchable 96 408 936 + 994176 upside-down 4375376 5867463425 542458256865 7792795138133 vampire 35164480 1299376408 1475812840 2970642136 wasteful 40 65 96 + 9999176 Zumkeller 40 96 176 + 99008 zygodrome 1133002200 113300220033 2222669966000 + 222266996600033