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Number

444 (four) is: 222^222

The only number which equals the number of letters in its name (four) when written in the English language

The only composite number nnn such that nnmidparenn−1!n \nmid \paren {n - 1}!nnmidparenn−1!

The only number nnn such that the alternating group AnA_nAn is not simple.

111st Term

The 111st semiprime: 4=2times24 = 2 \times 24=2times2

The 111st power of 444 after the zeroth 111: 4=414 = 4^14=41

The 111st Fermat pseudoprime to base 555: 54equiv5pmod45^4 \equiv 5 \pmod 454equiv5pmod4

The 111st nnn such that n!+1n! + 1n!+1 is square: see Brocard's Problem

The 111st positive integer nnn such that n−2kn - 2^kn−2k is prime for all kkk.

The 111st (with 121121121) of the 222 square numbers which are 444 less than a cube: 4=22=23−44 = 2^2 = 2^3 - 44=22=23−4

The 111st in the sequence formed by adding the squares of the first nnn primes: ds4=sumimathop=11pi2=22\ds 4 = \sum_{i \mathop = 1}^1 {p_i}^2 = 2^2ds4=sumimathop=11pi2=22

The 111st Smith number: 4=2+24 = 2 + 24=2+2

The smallest positive integer which can be expressed as the sum of 222 distinct lucky numbers in 111 different way: 4=1+34 = 1 + 34=1+3

The number of primes with no more than 111 digit: 222, 333, 555, 777

222nd Term

The 222nd square number: 4=2times2=22=1+34 = 2 \times 2 = 2^2 = 1 + 34=2times2=22=1+3

The first square of a prime number: 4=224 = 2^24=22

The 222nd square number after 111 to be the divisor sum value of some (strictly) positive integer: 4=mapsigma_134 = \map {\sigma_1} 34=mapsigma_13

The 222nd tetrahedral number after 111: 4=1+3=dfrac2paren2+1paren2+264 = 1 + 3 = \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 64=1+3=dfrac2paren2+1paren2+26

The 222nd after 111 of the 333 tetrahedral numbers which are also square

The 222nd trimorphic number after 111: 43=6mathbf44^3 = 6 \mathbf 443=6mathbf4

The 222nd power of 222 after (1)(1)(1), 222: 4=224 = 2^24=22

The 222nd powerful number after 111

The 222nd integer mmm after 333 such that m!−1m! - 1m!−1 (its factorial minus 111) is prime: 4!−1=24−1=234! - 1 = 24 - 1 = 234!−1=24−1=23

The 222nd integer nnn after 333 such that dsm=sumkmathop=0n−1paren−1kparenn−k!=n!−parenn−1!+parenn−2!−parenn−3!+cdotspm1\ds m = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1dsm=sumkmathop=0n−1paren−1kparenn−k!=n!−parenn−1!+parenn−2!−parenn−3!+cdotspm1 is prime: 4!−3!+2!−1!=174! - 3! + 2! - 1! = 174!−3!+2!−1!=17

The 222nd even number after 222 which cannot be expressed as the sum of 222 composite odd numbers.

The 222nd square after 111 which has no more than 222 distinct digits and does not end in 000: 4=224 = 2^24=22

The 222nd even integer after 222 that cannot be expressed as the sum of 222 prime numbers which are each one of a pair of twin primes.

The 222nd power of 222 after 111 which is the sum of distinct powers of 333: 4=22=30+314 = 2^2 = 3^0 + 3^14=22=30+31

The 222nd of 353535 integers less than 919191 to which 919191 itself is a Fermat pseudoprime: 333, 444, ldots\ldotsldots

The 222nd integer nnn after 333 for which the Ramanujan-Nagell equation x2+7=2nx^2 + 7 = 2^nx2+7=2n has an integral solution: 32+7=16=243^2 + 7 = 16 = 2^432+7=16=24

The number of different binary operations with an identity element that can be applied to a set with 222 elements

The total number of permutations of rrr objects from a set of 222 objects, where 1lerle21 \le r \le 21lerle2

333rd Term

The 333rd Lucas number after (2)(2)(2), 111, 333: 4=1+34 = 1 + 34=1+3

The 333rd highly composite number after 111, 222: mapsigma_04=3\map {\sigma_0} 4 = 3mapsigma_04=3

The 333rd superabundant number after 111, 222: dfracmapsigma_144=dfrac74=1cdotp75\dfrac {\map {\sigma_1} 4} 4 = \dfrac 7 4 = 1 \cdotp 75dfracmapsigma_144=dfrac74=1cdotp75

The 333rd almost perfect number after 111, 222: mapsigma_14=7=8−1\map {\sigma_1} 4 = 7 = 8 - 1mapsigma_14=7=8−1

The 333rd after 111, 222 of the 242424 positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

The 333rd of the 555 known powers of 222 whose digits are also all powers of 222: 111, 222, 444, ldots\ldotsldots

The 333rd positive integer after 222, 333 which cannot be expressed as the sum of distinct pentagonal numbers

444th Term

The 444th Ulam number after 111, 222, 333: 4=1+34 = 1 + 34=1+3

The 444th highly abundant number after 111, 222, 333: mapsigma_14=7\map {\sigma_1} 4 = 7mapsigma_14=7

The 444th (strictly) positive integer after 111, 222, 333 which cannot be expressed as the sum of exactly 555 non-zero squares.

The 444th integer after 000, 111, 222 which is palindromic in both decimal and ternary: 410=1134_{10} = 11_34_10=11_3

The 444th of the trivial 111-digit pluperfect digital invariants after 111, 222, 333: 41=44^1 = 441=4

The 444th positive integer after 111, 222, 333 such that all smaller positive integers coprime to it are prime

The 444th of the (trivial 111-digit) Zuckerman numbers after 111, 222, 333: 4=1times44 = 1 \times 44=1times4

The 444th of the (trivial 111-digit) harshad numbers after 111, 222, 333: 4=1times44 = 1 \times 44=1times4

The 444th (strictly) positive integer after 111, 222, 333 which cannot be expressed as the sum of distinct primes of the form 6n−16 n - 16n−1

The 444th after 111, 222, 333 of 212121 integers which can be represented as the sum of two primes in the maximum number of ways

555th Term

The 555th integer nnn after 000, 111, 222, 333 such that 2n2^n2n contains no zero in its decimal representation: 24=162^4 = 1624=16

The 555th integer nnn after 000, 111, 222, 333 such that 5n5^n5n contains no zero in its decimal representation: 54=6255^4 = 62554=625

The 555th integer nnn after 000, 111, 222, 333 such that both 2n2^n2n and 5n5^n5n have no zeroes: 24=162^4 = 1624=16, 54=6255^4 = 62554=625

The 555th number after 000, 111, 222, 333 which is (trivially) the sum of the increasing powers of its digits taken in order: 41=44^1 = 441=4

Miscellaneous

The number of faces and vertices of a tetrahedron

The number of sides and vertices of a square

The number of faces which meet at each vertex of a regular octahedron

The number of dimensions in Einstein's space-time

There exist exactly 444 Kepler-Poinsot polyhedra

Every positive integer can be expressed as the sum of at most 444 squares.

Arithmetic Functions on 444

| | | | | | | \(\ds \map {\sigma_0} { 4 }\) | \(=\) | | | | \(\ds 3\) | | | $\sigma_0$ of 444 | | | - | | | | | ----------------------------------- | ------- | | | | ------------ | | | --------------------------------------------------------------------------------- | | | | | | | | | \(\ds \map \phi { 4 }\) | \(=\) | | | | \(\ds 2\) | | | $\phi$ of 444 | | | | | | | | | \(\ds \map {\sigma_1} { 4 }\) | \(=\) | | | | \(\ds 7\) | | | $\sigma_1$ of 444 | |

Also see

Previous in Sequence: 111

Previous in Sequence: 222

Previous in Sequence: 333

Next in Sequence: 555

Next in Sequence: 666

Next in Sequence: 777

Next in Sequence: 999

Next in Sequence: 101010 and above

Historical Note

The number $4$, as was $8$, was associated by the Pythagoreans with the concept of justice, being evenly balanced: 4=2+24 = 2 + 24=2+2, where $2$ is the principle of diversity.

Throughout history, the number $4$ has been regarded with particular significance.

There were originally believed to be 444 elements out of which everything was formed:

Earth, Air, Fire and Water.

There are 444 humours:

Sanguine, Melacholic, Choleric and Phlegmatic.

There are 444 cardinal points of the compass:

North, East, South and West.

There are 444 seasons of the year:

Spring, Summer, Autumn and Winter.

In the Old Testament, there were 444 rivers which watered the Garden of Eden:

Pishon, Gihon, the Tigris, and the Euphrates.

In Islam, these rivers are:

Saihan (Syr Darya), Jaihan (Amu Darya), Furat (Euphrates) and Nil (Nile).

In the New Testament, there are 444 Gospels:

Matthew, Mark, Luke and John.

To bring it all full circle, there are believed to be 444 fundamental forces of nature:

Gravitation, the Weak Force, the Electromagnetic Force, the Strong Force.

The most aesthetically pleasing musical intervals are those whose frequencies are associated with the ratio of 1:41 : 41:4.

Linguistic Note

Words derived from or associated with the number 444 include:

quadruped: an animal with 444 feet

Sources