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pentagonal numbers
Pentagonal numbers are figurate numbers of the form  $n(3n-1)/2$  (see picture aside).

Every pentagonal number is equal to a triangular number divided by 3.

The sum of the reciprocals of pentagonal numbers is

\[\frac{1}{3} \left(9 \log(3)-\sqrt{3} \pi \right)\,.\]

The first pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715 more terms

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

A graph displaying how many pentagonal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of pentagonal numbers
Useful links Mathworld, Pentagonal Number
Wikipedia, Pentagonal number
OEIS, Sequence A000326
Pentagonal numbers can also be... (you may click on names or numbers and on + to get more values)

aban 12 22 35 + 998835000651 abundant 12 70 176 + 49971090 Achilles 16008300 34346991901 270990840125 admirable 12 70 532 + 44045632 alternating 12 70 92 + 963616901 amenable 12 92 117 + 999918232 amicable 991209208550 apocalyptic 247 287 925 + 29751 arithmetic 22 35 51 + 9998795 automorphic 376 binomial 35 70 210 + 9995127883720 brilliant 35 247 1247 + 993885751 c.decagonal 265651 61716605501 c.heptagonal 22 11572 265651 + 38881099834501 c.nonagonal 7906276 57722156241751 c.octagonal 9801 94109401 903638458801 c.pentagonal 51 3151 195301 + 178688837791801 c.square 145 925 28085 + 254088532692545 cake 176 47972 17862376 congruent 22 70 92 + 9983310 constructible 12 51 Cunningham 35 145 1001 + 127942295300965 Curzon 210 330 590 + 199578570 cyclic 35 51 145 + 9967837 D-number 51 d-powerful 376 3432 3725 + 9975572 de Polignac 17767 20827 31901 + 99825367 decagonal 12376 118837251 1141075274626 deceptive 4187 118301 250717 + 99145087427 deficient 22 35 51 + 9998795 dig.balanced 12 35 210 + 199890132 Duffinian 35 247 425 + 9991051 eban 4030 economical 35 145 287 + 19927215 emirpimes 51 1247 1717 + 96621001 equidigital 35 145 287 + 19927215 esthetic 12 210 3432 12345676565 Eulerian 247 evil 12 51 92 + 999840777 Friedman 2501 12105 32782 + 399126 frugal 585625 1572352 1628125 + 892222787 gapful 176 330 1001 + 99994534200 Gilda 330 happy 70 176 376 + 9890652 Harshad 12 70 117 + 9999553032 heptagonal 4347 16701685 64167869935 246532939589097 hex 7957 1185037 10596309577 1578130224697 hexagonal 40755 1533776805 57722156241751 highly composite 12 hoax 22 2625 4676 + 99923285 Hogben 651 5551 6247501 + 576009576039801 iban 12 22 70 + 777240 idoneal 12 22 70 + 330 inconsummate 1926 2147 3876 + 991047 interprime 12 176 376 + 99874320 Jordan-Polya 12 junction 117 210 715 + 99996755 katadrome 51 70 92 + 643210 Kynea 287 Lehmer 51 247 5551 + 999932033707 Leyland 145 lucky 51 477 651 + 9867555 Lynch-Bell 12 1926 magnanimous 12 70 92 + 1001 metadrome 12 35 145 + 1247 modest 3577 10045 15555 + 1793439837 Moran 117 247 1926 Motzkin 51 nialpdrome 22 51 70 + 8888842110 nonagonal 651 180868051 95317119801 nude 12 22 1335 + 444164292 O'Halloran 12 oban 12 35 70 + 925 octagonal 176 1575425 234631320 + 312461813932000 odious 22 35 70 + 999995690 palindromic 22 1001 2882 + 264571020175462 pancake 22 92 4187 + 25163130952051 panconsummate 12 pandigital 210 2882 16485 + 9120345876 partition 22 176 pernicious 12 22 35 + 9998795 Perrin 12 22 51 persistent 5329461870 10981754362 12087934465 + 98678243510 plaindrome 12 22 35 + 13444557 Poulet 7957 241001 1419607 + 999894027031901 power 9801 94109401 903638458801 powerful 9801 16008300 94109401 + 903638458801 practical 12 176 210 + 9983310 prim.abundant 12 70 532 + 69601610 primorial 210 pronic 12 210 116622 + 186039792969240 Proth 145 2625 43265 + 733006004225 pseudoperfect 12 176 210 + 998376 repdigit 22 repunit 651 2380 5551 + 576009576039801 Rhonda 1451892 Ruth-Aaron 715 1520 8855 + 509798229110 Sastry 715 self 176 782 1335 + 999918232 self-describing 22 semiprime 22 35 51 + 99898801 sliding 70 425 925 1001 Smith 22 852 27270 + 99923285 sphenic 70 590 651 + 99531901 square 9801 94109401 903638458801 straight-line 210 852 strobogrammatic 1001 super Niven 12 70 210 + 8400 super-d 247 715 782 + 9991051 superabundant 12 tau 12 376 852 + 995353520 tetrahedral 35 triangular 210 40755 7906276 + 57722156241751 trimorphic 51 376 uban 12 22 35 + 96000092000022 Ulam 145 1335 3290 + 9721901 undulating 1717 unprimeable 532 1335 1926 + 9983310 untouchable 210 782 852 + 986176 upside-down 852 5735 863742 346369147467 vampire 1128017682 1627054870 1710399852 + 5258884176 wasteful 12 22 51 + 9998795 weird 70 4030 29330 Zuckerman 12 1119312 1322112 Zumkeller 12 70 176 + 98176 zygodrome 22 8855 55777 + 88844444113322