hexagonal numbers

Hexagonal numbers are figurate numbers of the form $n(2n-1)$

(see picture aside).

Every hexagonal number is also triangular number.

The sum of the reciprocals of hexagonal numbers is $\log 4$ .

All the numbers greater than 130 can be written as the sum of 4 hexagonal numbers, and according to Duke and Schulze-Pillot, every number above a certain threshold is the sum of at most 3 hexagonal numbers. The last number that needs 4 addends is probably 146858.

The first hexagonal numbers are 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630 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 hexagonal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Hexagonal Number
Wikipedia, Hexagonal number
OEIS, Sequence A000384

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

aban 15 28 45 + 999949000753 abundant 66 120 276 + 49995000 Achilles 74024028 1024060098375 admirable 66 120 491536 523776 alternating 45 276 325 + 985036305 amenable 28 45 120 + 999916840 apocalyptic 861 1128 1653 + 29646 arithmetic 15 45 66 + 9997156 astonishing 15 Bell 15 binomial 15 28 45 + 9999997965180 brilliant 15 703 1891 + 980957071 c.decagonal 2467531 5130890585101 c.heptagonal 16322041 4145734153 c.nonagonal 28 91 190 + 999999818878690 c.octagonal 1225 1413721 1631432881 1882672131025 c.pentagonal 276 1891 88831 + 954928097357851 c.triangular 1891 26335 71156485 + 37298379237211 cake 15 378 Carmichael 561 8911 10585 + 921323712961 congruent 15 28 45 + 9988215 constructible 15 120 32640 2147450880 Cunningham 15 28 120 + 575598885856165 Curzon 153 378 561 + 199590210 cyclic 15 91 435 + 9934653 D-number 15 3003 d-powerful 153 378 1326 + 9792525 de Polignac 79003 93961 166753 + 99934453 decagonal 1540 1777555 2051297326 2367195337045 deceptive 91 703 8911 + 99851537521 deficient 15 45 91 + 9988215 dig.balanced 15 120 153 + 199310595 double fact. 15 Duffinian 325 1225 3655 + 9916831 eban 66 46056 66066 economical 15 1225 1431 + 19955403 emirpimes 15 1891 12403 + 97071211 equidigital 15 1225 1431 + 19955403 esthetic 45 5676765 Eulerian 66 120 evil 15 45 66 + 999916840 factorial 120 fibodiv 28 Friedman 153 46665 327645 + 857395 frugal 117855 130816 195625 + 774821295 gapful 120 190 231 + 99982069551 happy 28 91 190 + 9988215 harmonic 28 496 8128 + 137438691328 Harshad 45 120 153 + 9999171820 heptagonal 121771 12625478965 hex 91 8911 873181 + 80505887094361 highly composite 120 2162160 hoax 861 9180 11476 + 99793128 Hogben 91 703 104653 + 160829915470003 house 7759830 hyperperfect 28 325 496 + 137438691328 iban 120 703 1770 + 717003 idoneal 15 28 45 + 190 inconsummate 276 630 861 + 976503 interprime 15 45 120 + 99906180 Jordan-Polya 120 junction 2016 3003 6216 + 99595441 Kaprekar 45 703 4950 + 49999995000000 katadrome 91 630 861 + 86320 Lehmer 15 91 435 + 999406030321 lonely 120 Lucas 5778 lucky 15 231 2415 + 9988215 Lynch-Bell 15 1326 41328 magic 15 metadrome 15 28 45 378 modest 10011 30135 84666 + 1896263736 Moran 45 153 190 narcissistic 153 nialpdrome 66 91 630 + 87777776521 nonagonal 325 5330229625 1353857339341 nude 15 66 1128 + 481228776 oban 15 28 66 + 780 octagonal 11781 113123361 1086210502741 odious 28 91 276 + 999827403 palindromic 66 3003 5995 + 636188414881636 panconsummate 15 45 91 231 pandigital 15 120 8385 + 9654871320 partition 15 231 pentagonal 40755 1533776805 57722156241751 perfect 28 496 8128 + 137438691328 pernicious 28 66 91 + 9925740 persistent 1520843976 2718609453 3574816290 + 97486512903 plaindrome 15 28 45 + 355555577777778 Poulet 561 2701 4371 + 999961448323753 power 1225 1413721 1631432881 1882672131025 powerful 1225 1413721 74024028 + 194838725161125 practical 28 66 120 + 9961416 prim.abundant 66 21945 30628 + 53804751 pronic 7140 8239770 9508687656 10973017315470 pseudoperfect 28 66 120 + 33550336 repdigit 66 repfigit 28 repunit 15 91 703 + 160829915470003 Ruth-Aaron 15 153 4186 + 921236838753 Saint-Exupery 780 25743900 self 378 435 703 + 999112051 semiprime 15 91 703 + 97071211 Smith 378 861 2556 + 99906180 sphenic 66 190 231 + 99934453 square 1225 1413721 1631432881 1882672131025 star 49141 1849384153 69599723176621 straight-line 630 super Niven 120 630 2016 + 2031120 super-d 190 231 1431 + 9916831 superabundant 120 2162160 tau 276 2016 2556 + 996342480 taxicab 61249825000 tetrahedral 120 1540 7140 tetranacci 15 triangular 15 28 45 + 999999997764120 uban 15 28 45 + 98000077000015 Ulam 28 861 1035 + 9827961 undulating 5151 unprimeable 325 3486 4186 + 9997156 untouchable 120 276 1128 + 990528 upside-down 28 91 485628284526 vampire 14340690 1820789685 2510959545 wasteful 28 45 66 + 9997156 Zuckerman 15 382192128 911111328 Zumkeller 28 66 120 + 98346 zygodrome 66 1188111 2200066611 + 558822550004400