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centered octagonal numbers
Centered octagonal numbers are figurate numbers (see picture aside) defined as  $O_n=4n(n-1)+1$. They coincide with odd squares.

Three formulas involving centered octagonal numbers:

\[\sum_{n=1}^{\infty}\frac{1}{O_n}=\frac{\pi^2}{8}\,,\quad \sum_{n=0}^{\infty}\frac{O_n}{n!}=5e\,, \quad\mbox{and}\quad\sum_{n=1}^{\infty}\frac{O_n}{2^n}=17\,. \]

The first centered octagonal numbers are 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521 more terms

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

A graph displaying how many centered octagonal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of centered octagonal numbers
Useful links Wikipedia, Centered octagonal number
OEIS, Sequence A016754
Centered octagonal numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 81 15625 59049 + 8741371209561 aban 25 49 81 + 999998000001 abundant 11025 99225 245025 + 49491225 alternating 25 49 81 + 987656329 amenable 25 49 81 + 999887641 apocalyptic 361 529 841 + 29929 arithmetic 49 169 361 + 9853321 automorphic 25 625 8212890625 binomial 1225 1413721 1631432881 1882672131025 brilliant 25 49 121 + 999002449 c.decagonal 361 116281 37442161 + 3882078149401 c.heptagonal 841 755161 678133681 + 546848356574521 c.nonagonal 1225 1413721 1631432881 1882672131025 c.pentagonal 22801 1666681 47411143081 3465632747641 c.square 25 841 28561 + 43892069261881 c.triangular 361 6241 3463321 + 319312633001041 canyon 529 625 729 + 865124569 cube 729 15625 117649 + 976929722015625 Cullen 25 Curzon 81 441 729 + 198218241 d-powerful 4225 5329 49729 + 8323225 de Polignac 40401 62001 96721 + 99620361 deceptive 237169 deficient 25 49 81 + 9991921 dig.balanced 49 169 225 + 199967881 Duffinian 25 49 81 + 9991921 economical 25 49 81 + 19989841 emirpimes 49 169 289 + 99460729 enlightened 3515625 387420489 3486784401 + 137858491849 equidigital 25 49 81 + 19936225 eRAP 265225 616225 13213225 + 519926081481 esthetic 121 12321 1234321 + 123456787654321 evil 169 225 441 + 999255321 Friedman 25 121 289 + 990025 frugal 625 729 1369 + 999887641 gapful 121 225 2025 + 99965365929 Gilda 49 happy 49 1521 4225 + 9966649 Harshad 81 225 441 + 9995000625 heptagonal 81 5929 2307361 + 350709705290025 hex 169 32761 6355441 + 46399815451081 hexagonal 1225 1413721 1631432881 1882672131025 hoax 361 2401 39601 + 90611361 iban 121 441 2401 + 741321 idoneal 25 inconsummate 441 1521 2601 + 962361 interprime 81 225 441 + 98545329 junction 2025 5625 7225 + 99860049 katadrome 81 841 961 lucky 25 49 169 + 9941409 magnanimous 25 49 625 + 22801 metadrome 25 49 169 + 134689 modest 49 82369 312481 + 1666027489 mountain 121 361 1521 + 123456787654321 nialpdrome 81 441 841 + 999887641 nonagonal 1089 8281 978121 + 708304623404049 oban 25 529 625 729 octagonal 225 43681 8473921 + 61866420601441 odious 25 49 81 + 999887641 palindromic 121 10201 12321 + 900075181570009 pancake 121 529 139129 + 937385441796001 panconsummate 81 121 169 361 pandigital 225 38025 314721 + 9614783025 pentagonal 9801 94109401 903638458801 pernicious 25 49 81 + 9991921 persistent 1532487609 3285697041 5102673489 + 96438197025 plaindrome 25 49 169 + 277777788888889 Poulet 1194649 12327121 power 25 49 81 + 49999988518489 powerful 25 49 81 + 999999898700625 prim.abundant 342225 Proth 25 49 81 + 274878955521 pseudoperfect 11025 99225 245025 + 893025 repunit 121 Rhonda 15625 342225 115154361 5547121441 Ruth-Aaron 25 49 1521 + 737521981681 self 121 6889 9409 + 999887641 semiprime 25 49 121 + 99460729 sliding 25 Smith 121 729 10201 + 96530625 square 25 49 81 + 999999898700625 star 121 11881 1164241 + 107341182792481 strobogrammatic 6889 69169 109181601 super-d 81 169 1369 + 9979281 tau 225 441 625 + 998117649 taxicab 49991146569 8936757492481 18476474674329 + 781111665140625 triangular 1225 1413721 1631432881 1882672131025 tribonacci 81 10609 trimorphic 25 49 625 + 8212890625 uban 25 49 81 + 98010099000025 Ulam 441 2809 3481 + 9690769 undulating 121 unprimeable 625 27225 30625 + 9579025 upside-down 926654481 vampire 19847025 51480625 1116695889 1480325625 wasteful 225 441 1089 + 9966649