Welcome to the most comprehensive review of the number 30 ever created. The number 30 possesses remarkable attributes,
including–and perhaps most profoundly–its role (along with its prime factors 2, 3 and 5) as a primary organizing principle
in the distribution of prime numbers. Before getting to that, here's a list of mathematical properties and other interesting facts
relating to this integer:
 30 is a natural number (aka whole number; counting number; nonnegative integer; positive integer) following 29 and
preceding 31.
 30 is a positive whole number with two digits.

30 is the 19th composite number (aka
nonprime number; nonnegative nonprime; natural nonprime; whole nonprime; counting nonprime).
 30 is an even number.
 30 is equal to: 2 + 4 + 6 + 8 + 10; 4 + 5 + 6 + 7 + 8; 2 + 3 + 5 + 8 + 12; 6 + 7 + 8 + 9; and 1 + 4 + 9 + 16.

30 is divisible by 1, 2, 3, 5, 6, 10, 15 and 30; its divisors sum to 72; it is thus congruent
to 2 mod 4: a(n) = 4n+2, i.e., is an integer possessing equal numbers of odd and even divisors (aka
singly even numbers).

30 is the 3rd primorial (2∗3∗5);
the first two being 2 (1∗2) and 6 (2∗3). It follows that all primorials ≥ 30 are evenly divisible by 30.
 30 is the smallest sphenic number; divisible by exactly
3 distinct primes (2, 3 and 5) and a total of 8 divisors (see bullet, directly above).
 30 is the smallest of the four terms in the arithmetic progression of positive integers possessing three distinct prime factors:
3066102138.
 30 is smallest number not the sum of three integer cubes.
 30 is equal to multiples of the first three prime numbers, namely:
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 15∗2 = 30
3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 10∗3 = 30
5 + 5 + 5 + 5 + 5 + 5 = 6∗5 = 30
 30 is a square pyramidal number
in that it is the sum of the first four
square numbers (1² + 2² + 3² + 4² = 30), as illustrated below:
 30 is an octagonal pyramidal number: n(n+1)(2n1)/2.
 30 is a 5smooth number (aka Hamming sequence) in that its prime divisors are all ≤ 5.
 30 is a 7smooth number (aka highly composite number or humble number) in that its prime divisors
are all ≤ 7.
 30 is an 11smooth number in that its prime divisors are all ≤ 11.
 30 is the product of the first 5 nonzero Fibonacci numbers (F_{1}, ..., F_{5}),
thus: 1∗1∗2∗3∗5 = 30.
 30 is equal to (1+5)+(2+5)+(3+5)+(4+5) = 6+7+8+9 = 30
 30 is equal to 1^{2} + 2^{2} + 5^{2}.
 30 is equal to 2^{5}2 and n^{5}n is divisible by 30.
 30 is a pronic number
(aka oblong, rectangular or heteromecic number) and as such is the sum of two triangular numbers.
 30 is the smallest Giuga number.
 30 is the sum of two successive primes: 13+17=30.
 30 is the index number for Fibonacci number, F_{30} = 832040 = 5∗11∗23∗31∗61 = the largest
Fibonacci number with 6 digits.
 30 is the smallest integer with three distinct prime divisors.

30 "splits" twin primes
29 and 31, the latter both a Mersenne prime and a
lucky number. This relationship can also be expressed as
30 = average of twin primes 29 and 31 or as

30 is an abundant number
(aka excessive number) in that the sum of its divisors (72) exceeds 2n. 30's abundance is
thus calculated as: 72  (2∗30) = 12.
 30 is a highly abundant number.
 30 is a member of the Lower Wythoff sequence.

30 is the largest number such that all smaller numbers prime relative to it are actually 1 or prime; those numbers
being: 1, 7, 11, 13, 17, 19, 23 & 29. 30 is thus the largest member of the 10element sequence of
very round numbers (the other 9 elements in this sequence being 1, 2, 3, 4, 6, 8, 12, 18 & 24)
a reduced residue system consisting of only primes and 1. In 30's case the reduced residue is expressed as {30,{1, 7, 11, 13, 17, 19, 23, 29}}.
 30 is the number of ways to
color a cube with 6 colors.
 30 has an aliquot sum of 42.
 30 is a semiperfect number
(aka pseudoperfect number) in that a subset of its divisors (5, 10 and 15) = 30.

30 is the total unit length of the straight line segments used to construct a
star polygon
notated as {p/q} where p=10 and q=3 (q is termed the density of the star polygon; in this case the straight lines connect every 3rd point
of 10 equally spaced points lying on a circle's circumference.); thus {10/3}. The
SchlĂ¤fli symbol for this polygon is {30}. Illustration, below:
 30 is the number of sides possessed by the
triacontagon (a polygon).
 30 is the maximum number of edges that may be had by a regular
convex polyhedron (aka convex polytope).
 30 is the number of edges possessed by two of the five
Platonic solids:
the icosahedron
and the dodecahedron.
 30 is the number of identical vertices possessed by the
icosidodecahedron,
an Archimedean solid.
 30 is the number of square faces (along with 20 triangular and 12 pentagonal faces) used to construct the symmetrical Archimedean solid called a
rhombicosidodecahedron.
 30 is the area and perimeter of one of only two
Pythagorean triangles
whose areas equal their perimeters (51213 and 6810). This can also be termed: 30 is the
ordered area and ordered perimeter
of a primitive Pythagorean triangle.
 30 is the number of isohedra.
 30 is the smallest acute angle in a right triangle whose hypotenuse is twice as long as one of its sides.
 30 is the 15th member of a sequence quantifying the number of polygon edges constructible
with ruler and compass.
 30 is a twice hexagonal number of the form 2n(2n1) where n=3.
 30 is a squarefree integer; not divisible by
a square greater than 1, i.e. all of its prime factors are distinct.
 30 is a cubefree integer; not divisible by
any cube > 1.
 30 is a golden3 almost prime given that 2∗3∗5=30
and that the products of both 2∗3=6 and 3∗5=15 are golden semiprimes.
 30 is a 11gonal number (aka hendecagonal number)
of the form n(9n7)/2.
 30 is the 20th partial sum of the Kolakoski sequence.
 30 is a number n such that the sum of its digits squared is a square: 30^{2} = 900 = 9+0+0= 9 or 3^{2}.
 30 is the Least Common Multiple (LCM)
of the set 2, 3 and 5 making it the LCM of its own prime factors.
 30 and its multiples (60, 90, 120 ... n) are the only natural numbers divisible by 2, 3 and 5.
 30 is the sum of the 32nd Fibonacci number's digits; F_{32} = 2178309; 2+1+7+8+3+0+9=30.

30 is the Coxeter Group number h,
dual Coxeter number and the highest degree of fundamental invariance of the Lie Group
E_{8}. You'll note, looking
at the graphical representation of E_{8} (see image at top of this page) that the perimeters of every one of its
multiple concentric circles possesses 30 points. E_{8} has 2, 3 and 5torsion, and
its exponents are the coprimes up to 30, i.e., 1, 7, 11, 13, 17, 19, 23, and 29.
The graphic below superimposes an image of E_{8} with a star polygon and the 8 radii of a modulo 30 factorization wheel:
The Number 30 in Lie Group E_{8}: This graphic superimposes images of the star
polygon, modulo 30 wheel factorization radii and "E_{8} graph of the Gosset 421 polytope as a 2dimensional skew orthogonal projection
inside Petrie polygon ... an emulation of the hand drawn original by Peter McMullen" licensed by Creative Commons;
license terms here.
 30 is the number of integer partitions of
the number 9: P(9)=30.
 30 has 296 partitions into distinct parts: P(30)=296.
 30 in binary form: 11110_{2}.
 30 is an evil number, mathematically speaking, in that
it has an even number of 1's in its binary expansion.
 30 is a dopey number because its binary representation ends in an odd number of zeros.
 30 is a Harshad number (aka Niven number) in that it's
divisible by the sum of its digits.
 30 is a practical number.
 30 is an admirable number.
 30 is a balanced number.
 30 is a swinging factorial.
 30 is a polite number, i.e. is the sum of two or more
consecutive integers, namely: 4 + 5 + 6 + 7 + 8 = 30.
 30 is a skinny number in that when squared using "long division" there are no carries.
 30 is a single number (aka isolated number).
 30 is a sloping binary number.
 30 is a Zumkeller number.
 30 is a permutational number.
 30 is an idoneal number (aka suitable or convenient number) after Euler's "numerus idoneus".
 30 is a number having no prime gaps in its factorization.
 30 is a triangular matchstick number: 3n(n+1)/2; 3∗4(4+1)/2=30
 30 is a number of the form a^{2} + 5b^{2} with a and b as positive integers.
Thus, when a = 5 and b = 1: 5^{2} + 5(1^{2}) = 30.
 30 is the 13th value of the sequence: numbers that are not powers of primes.
 30 is the 3rd value in the sequence: 1^{n} + 2^{n} + 5^{n}.
 30 is the 3rd value for n in this sequence: n!/24.
 30 is the 4th value in this sequence (n=3): n^{3}+n; 3^{3}+3=30
 30 is the 4th value in this sequence (n=3): n(n+1)(n+2)/2; 3(3+1)(3+2)/2=30.
 30 is the 4th value in this sequence (n=3): n(n+7); 3(3+7)=30.
 30 is the 4th value in this sequence (n=3): (n/2)∗(3∗n + 11); (3/2)∗(3∗3+11)=30.
 30 is a number such that the number of its prime factors counted with multiplicity (3) is prime.
 30 is a number such that the sum of its prime factors counted with multiplicity (2 + 3 + 5 = 10) is not prime.
 30 is a number n such that primorial(n)/2 + 16 is prime.
 30 is a number n such that primorial(n)/2  16 is prime.
 30 is a number n such that n^{2}+(n1) is prime; 30^{2}+(301)=929 (prime).
 30 is a number n such that n^{2}+(n+1)^{2} is prime; 30^{2}+(30+1)^{2}=1861 (prime).
 30 is a number n such that n!+2/2 is prime.
 30 is a number n such that 2^{(n+1)} 1 is prime, thus: 2^{(30+1)} 1 = 2,147,483,647 (prime).
 30 is a number n such that 6n1 and 6n+1 are twin primes.
 30 is a number n such that 2n1 is prime; 2∗301=59.
 30 is a number n such that 2n+1 is prime; 2∗30+1=61.
 30 is a number n such that n^{3}+n+1 is prime.
 30 is a number of the form x^{2} + y^{2} + z^{2}, where x, y & z are ≥ 0
(and in this case 1^{2} + 2^{2} + 5^{2} = 30).
 30 is a 4dimensional figurate number.
 30 is a hypotenuse number in that its square is the sum of 2 distinct nonzero squares:
18^{2} + 24^{2} = 30^{2} or 324 + 576 = 900.
 30° = π/6 radians = (π/6)^{r}.
 30 has a digital root (aka repeated digital sum) of 3: 3+0=3. And, its
digit sum also equals 3.
 30 has five 1's in its baseφ representation.

The number 30, when plugged into
Euler's totient function, phi(n): phi(30)= 8, with the 8 integers smaller than and having no factors in common with 30 being:
1, 7, 11, 13, 17, 19, 23 & 29. Thirty is the largest integer with this property.
 Modulo 30 of all prime numbers (with the exception of 2, 3 and 5) must be 1, 7, 11, 13, 17, 19, 23 or 29.
 Modulo 30's quadratic residues (making them congruent
with perfect squares) are 1 and 19.
 Below is a magic square where all horizontal, vertical and principal
(cornertocorner) diagonal sums total 30 and the horizontal and vertical sums combined = 360, making it, with 36 squares, a perfect
square of the first perfect number (6) squared in terms of number of squares, and when multiplied by 10 we have a perfect square
representing a circle. Explore its beautiful symmetries involving 10's derived from digit sum (aka Vedic)
arithmetic, the significance of which is explored under our topic,
Factorization Utilizing Digital Root Sequencing.
 Here's another magic square with rows, columns and principal diagonals totalling 30; in this case the first three prime numbers {2,3,5} are
configured into 3 square matrices (each containing three 2's, 3's and 5's) which in turn are multiplied times three to make magic (created by Gary Croft
on 25 April, 2012):
 When arrayed in eight columns, the set of all natural numbers not divisible by 2, 3 and 5 (which by definition
consists of 1 and all prime numbers >5 and their multiples) possesses perfect symmetry involving the number 30, as illustrated below. [Also note that
the sum of the digital root sums (1 + 7 + 2 + 4 + 8 +1 + 5 + 2) of the first 8 elements in this set (1, 7, 11, 13, 17, 19, 23, 29) = 30.]:
 When twin primes and twin prime candidates
≥ [11, 13] are arrayed in this divergent (aka harmonic) sequence: 11{+2+4+2+10+2+10} {repeat ... n}, the
intervals between them (2+4+2+10+2+10) total 30, as shown in the matrix below (For more on this subject
click here.):
TWIN PRIME DISTRIBUTION MATRIX
11 {+2+4+2+10+2+10} {repeat ... ∞}
11 
+2 
13 

+4 

17 
+2 
19 

+10 

29 
+2 
31 

+10 
41 
+2 
43 

+4 

47 
+2 
49 

+10 

59 
+2 
61 

+10 
71 
+2 
73 

+4 

77 
+2 
79 

+10 

89 
+2 
91 

+10 
101 
+2 
103 

+4 

107 
+2 
109 

+10 

119 
+2 
121 

+10 
131 
+2 
133 

+4 

137 
+2 
139 

+10 

149 
+2 
151 

+10 
161 
+2 
163 

+4 

167 
+2 
169 

+10 

179 
+2 
181 

+10 
191 
+2 
193 

+4 

197 
+2 
199 

+10 

209 
+2 
211 

+10 
221 
+2 
223 

+4 

227 
+2 
229 

+10 

239 
+2 
241 

+10 
251 
+2 
253 

+4 

257 
+2 
259 

+10 

269 
+2 
271 

+10 
281 
+2 
283 

+4 

287 
+2 
289 

+10 

299 
+2 
301 

+10 
311 
+2 
313 

+4 

317 
+2 
319 

+10 

329 
+2 
331 

+10 
341 
+2 
343 

+4 

347 
+2 
349 

+10 

359 
+2 
361 

+10 
371 
+2 
373 

+4 

377 
+2 
379 

+10 

389 
+2 
391 

+10 
401 
+2 
403 

+4 

407 
+2 
409 

+10 

419 
+2 
421 

+10 
431 
+2 
433 

+4 

437 
+2 
439 

+10 

449 
+2 
451 

+10 
461 
+2 
463 

+4 

467 
+2 
469 

+10 

479 
+2 
481 

+10 
491 
+2 
493 

+4 

497 
+2 
499 

+10 

509 
+2 
511 

+10 
521 
+2 
523 

+4 

527 
+2 
529 

+10 

539 
+2 
541 

+10 
551 
+2 
553 

+4 

557 
+2 
559 

+10 

569 
+2 
571 

+10 
= Prime 
= Interval 
= Composite 
prime 
+2 
prime 
= Prime Pair 
 30 is interesting when contextualized within the set of natural numbers 1100 (Hint: Start with the bottom row, then work up.):
 30 is often cited as the "sweet spot" for optimizing sample size.
 30 is the distance in degrees between the numbers on a clock.
 30 is the total number of major and minor keys in Western tonal music, including enharmonic equivalents.
 30 is the number of upright stones that originally encompassed the
Sarsen Circle, Stonehenge's best known feature.