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tetrahedral numbers
Tetrahedral numbers are defined as  $Te_n=n(n+1)(n+2)/6$  and are among the simplest tridimensional figurate numbers (see picture aside).

By construction,  $Te_n = \sum_{k=1}^n T_k$, where  $T_k$  is the  $k$-th triangular number and it also holds  $Te_{2n}=4\cdot Te_n+4\cdot Te_{n-1}$.

The sum of the reciprocals of tetrahedral number is 3/2 and we also have

\[ \sum_{k=1}^\infty\frac{Te_k}{2^k}=8,\quad\sum_{k=1}^\infty\frac{Te_k}{k!}=\frac{13e}{6},\quad\mathrm{and\ }\sum_{k=1}^\infty\frac{1}{Te_k+T_k}=\frac{163}{200}. \]

The first tetrahedral numbers are 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540 more terms

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

A graph displaying how many tetrahedral numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of tetrahedral numbers
Useful links Mathworld, Tetrahedral Number
Wikipedia, Tetrahedral number
OEIS, Sequence A000292
Tetrahedral numbers can also be... (you may click on names or numbers and on + to get more values)

aban 10 20 35 + 702569000685 abundant 20 56 84 + 49902940 admirable 20 56 84 + 349504 alternating 10 56 165 + 890169874 amenable 20 56 84 + 999800616 amicable 220 17296 9363584 apocalyptic 220 286 1140 + 29260 arithmetic 20 35 56 + 9962680 betrothed 2024 binomial 10 20 35 + 9999481661774 brilliant 10 35 c.nonagonal 10 1540 c.triangular 10 4960 428536 canyon 816 969 2024 + 6435689 congruent 20 56 84 + 9962680 constructible 10 20 120 + 46912496107520 Cunningham 10 35 120 + 435730689800 Curzon 165 2925 3654 + 192937325 cyclic 35 455 9139 + 9886435 d-powerful 366145 562475 657359 + 6435689 de Polignac 302621 366145 657359 + 56677949 decagonal 10 1540 10660 deficient 10 35 165 + 9886435 dig.balanced 10 35 56 + 196821856 Duffinian 35 1771 6545 + 9290431 eban 56 4060 6044060 economical 10 35 12341 + 19131795 equidigital 10 35 12341 + 19131795 eRAP 20 esthetic 10 56 5456 6545 Eulerian 120 evil 10 20 120 + 989928820 factorial 120 fibodiv 969 Friedman 45760 gapful 120 165 220 + 99668626200 Gilda 220 364 happy 10 680 1330 + 9962680 harmonic 695520 Harshad 10 20 84 + 9993352005 heptagonal 286 hexagonal 120 1540 7140 highly composite 120 27720 7207200 hoax 84 364 26235 + 91558740 iban 10 20 120 + 410040 idoneal 10 120 165 inconsummate 84 816 3654 + 924176 interprime 56 120 165 + 91223769 Jordan-Polya 120 junction 816 2925 16215 + 99846044 katadrome 10 20 84 85320 Lehmer 9139 1004731 10507399 + 871058910085 lonely 120 lucky 1771 4495 9139 + 9886435 Lynch-Bell 816 3276 magic 2925 magnanimous 20 56 metadrome 35 56 modest 3654 4060 273819 + 22108415 Moran 84 mountain 120 165 286 + 12347930 nialpdrome 10 20 84 + 85333333320000 nonagonal 969 234136 nude 816 1771 3276 + 12614424 O'Halloran 20 84 oban 10 20 35 + 969 octagonal 560 20337240 odious 35 56 84 + 999800616 palindromic 969 1771 6378736 pancake 56 1771 panconsummate 10 20 pandigital 120 37820 43680 + 131908700 partition 56 pentagonal 35 pernicious 10 20 35 + 9810580 Perrin 10 persistent 18259738640 26507314869 plaindrome 35 56 455 Poulet 6891657409 777080801185 1581289265305 + 674228077807189 power 19600 powerful 19600 practical 20 56 84 + 9962680 prim.abundant 20 56 364 pronic 20 56 7140 1414910 pseudoperfect 20 56 84 + 988260 repunit 364 Ruth-Aaron 1330 2300 2600 + 48134020 Saint-Exupery 497640 130459630080 192805230220200 self 20 165 10660 + 994856555 semiprime 10 35 sliding 20 1330 Smith 4960 9880 37820 + 95988256 sphenic 165 286 455 + 92568571 square 19600 super Niven 10 20 120 + 7207200 super-d 969 9880 10660 + 9290431 superabundant 120 27720 7207200 tau 56 84 560 + 986652720 tetranacci 56 triangular 10 120 1540 7140 uban 10 20 35 56 Ulam 14190 19600 50116 + 9660036 undulating 969 unprimeable 1140 1330 4060 + 9886435 untouchable 120 2024 2600 + 988260 vampire 1563429820 wasteful 20 56 84 + 9962680 Zuckerman 816 3276 Zumkeller 20 56 84 + 98770 zygodrome 55228866 1188887700