3

Number
$3$ (three) is:


 * The $2$nd prime number after $2$


 * The only Fibonacci prime whose index is composite

$1$st Term

 * The $1$st odd prime


 * The $1$st Proth prime:
 * $3 = 1 \times 2^1 + 1$


 * The smaller element of the $1$st pair of twin primes, with $5$


 * The $1$st power of $3$ after the zeroth $1$:
 * $3 = 3^1$


 * The $1$st prime number of the form $4 n + 3$:
 * $3 = 4 \times 0 + 3$


 * The $1$st Mersenne number and Mersenne prime:
 * $3 = 2^2 - 1$


 * The $1$st Fermat number and Fermat prime:
 * $3 = 2^{\paren {2^0} } + 1$


 * The $1$st prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
 * $3 \# - 1 = 2 \times 3 - 1 = 5$


 * The $1$st of an arithmetic sequence of primes:
 * $3$, $5$, $7$


 * The $1$st prime number $p$ which satisfies the equation $p^2 \divides \paren {10^p - 10}$:
 * $3^2 \divides \paren {10^3 - 10}$


 * The $1$st integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
 * $3! - 1 = 6 - 1 = 5$


 * The $1$st of $3$ primes of the form $2 x^2 + 3$:
 * $2 \times 0^2 + 3 = 3$


 * The $1$st Euclid number after the zeroth $2$:
 * $3 = p_1\# + 1 = 2 + 1$


 * The $1$st integer whose square is the sum of $2$ coprime cubes:
 * $3^2 = 2^3 + 1^3$


 * The $1$st prime number that can be found starting from the beginning of the decimal expansion of $\pi$ (pi):
 * $3 (\cdotp 14159 \ldots)$


 * The $1$st of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime


 * The $1$st integer $n$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
 * $3! - 2! + 1! = 5$


 * The $1$st prime number which is palindromic in both decimal and binary:
 * $3_{10} = 11_2$


 * The smallest positive integer the decimal expansion of whose reciprocal has a period of $1$:
 * $\dfrac 1 3 = 0 \cdotp \dot 3$

$2$nd Term

 * The $2$nd (trivial, $1$-digit, after $2$) palindromic prime


 * The $2$nd Sophie Germain prime after $2$:
 * $2 \times 3 + 1 = 7$, which is prime


 * The $2$nd triangular number after $1$:
 * $3 = 1 + 2 = \dfrac {2 \paren {2 + 1} } 2$


 * The index of the $2$nd Mersenne prime after $2$:
 * $M_3 = 2^3 - 1 = 7$


 * The $2$nd Fibonacci prime after $2$


 * The $2$nd permutable prime after $2$


 * The $2$nd Lucas number after $(2)$, $1$:
 * $3 = 2 + 1$


 * The $2$nd Lucas prime after $2$


 * The $2$nd lucky number after $1$:
 * $1$, $3$, $\ldots$


 * The $2$nd palindromic lucky number after $1$:
 * $1$, $3$, $\ldots$


 * The $2$nd Stern number after $1$


 * The $2$nd Cullen number after $1$:
 * $3 = 1 \times 2^1 + 1$


 * The $2$nd term of Göbel's sequence after $(1)$, $2$:
 * $3 = \paren {1 + 1^2 + 2^2} / 2$


 * The $2$nd (trivially) two-sided prime after $2$


 * The $2$nd prime number after $2$ consisting (trivially) of a string of consecutive ascending digits


 * The $2$nd prime $p$ after $2$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
 * $3 \# + 1 = 2 \times 3 + 1 = 7$


 * The $2$nd after $2$ of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
 * $\dfrac {3^3 - 1} {3 - 1} = 13$


 * The sum of the first $2$ factorials:
 * $3 = 1! + 2!$


 * The index (after $2$) of the $2$nd Woodall prime:
 * $3 \times 2^3 - 1 = 23$


 * The $2$nd number after $1$ whose divisor sum is square:


 * The $2$nd of the lucky numbers of Euler after $2$:
 * $n^2 + n + 3$ is prime for $0 \le n < 2$


 * The $2$nd positive integer after $1$ of which the product of its Euler $\phi$ function and its divisor count equals its divisor sum:
 * $\map \phi 3 \map {\sigma_0} 3 = 2 \times 2 = 4 = \map {\sigma_1} 3$


 * The $2$nd positive integer solution after $1$ to $\map \phi n = \map \phi {n + 1}$:
 * $\map \phi 3 = 2 = \map \phi 4$


 * The $2$nd positive integer which is not the sum of $1$ or more distinct squares:
 * $2$, $3$, $\ldots$


 * The $2$nd element of the Fermat set after $1$


 * The $2$nd positive integer after $2$ which cannot be expressed as the sum of distinct pentagonal numbers


 * The $2$nd integer $n$ after $1$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
 * $\map {\sigma_0} 3 = 2$, $\map \phi 3 = 2$, $\map {\sigma_1} 3 = 4$


 * The $2$nd of the $2$nd pair of consecutive integers whose product is a primorial:
 * $2 \times 3 = 6 = 3 \#$


 * The $2$nd prime number after $2$ to be of the form $n! + 1$ for integer $n$:
 * $2! + 1 = 2 + 1 = 3$
 * where $n!$ denotes $n$ factorial


 * The $2$nd Lucas number after $1$ which is also triangular:
 * $3 = \ds \sum_{k \mathop = 1}^2 k = \dfrac {2 \times \paren {2 + 1} } 2 = 2 + 1$


 * The $2$nd Euclid prime after $2$:
 * $3 = p_1\# + 1 = 2 + 1$


 * The $2$nd odd positive integer after $1$ such that all smaller odd integers greater than $1$ which are coprime to it are prime


 * The $2$nd odd positive integer after $1$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

$3$rd Term

 * The $3$rd palindromic triangular number after $0$, $1$ whose index is itself palindromic:
 * $3 = T_2$


 * The $3$rd highly abundant number after $1$, $2$:
 * $\map {\sigma_1} 3 = 4$


 * The $3$rd Ulam number after $1$, $2$:
 * $3 = 1 + 2$


 * The $3$rd positive integer after $1$, $2$ such that all smaller positive integers coprime to it are prime


 * The $3$rd integer after $0$, $1$ which is palindromic in both decimal and binary:
 * $3_{10} = 11_2$


 * The $3$rd (strictly) positive integer after $1$, $2$ which cannot be expressed as the sum of exactly $5$ non-zero squares


 * The $3$rd after $0$, $1$ of the $5$ Fibonacci numbers which are also triangular


 * The $3$rd palindromic triangular number after $0$, $1$


 * The $3$rd of the trivial $1$-digit pluperfect digital invariants after $1$, $2$:
 * $3^1 = 3$


 * The $3$rd of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$:
 * $3 = 1 \times 3$


 * The $3$rd of the (trivial $1$-digit) harshad numbers after $1$, $2$:
 * $3 = 1 \times 3$


 * The $3$rd Ramanujan-Nagell number after $0$, $1$:
 * $3 = 2^2 - 1 = \dfrac {2 \paren {2 + 1} } 2$


 * The $3$rd (strictly) positive integer after $1$, $2$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$


 * The $3$rd after $1$, $2$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways


 * The $3$rd after $0$, $2$ of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
 * $1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 225 = 15^2$


 * The index of the $3$rd Mersenne number after $1$, $2$ which asserted to be prime


 * The number of different representations of $1$ as the sum of $3$ unit fractions

$4$th Term

 * The $4$th Fibonacci number after $1$, $1$, $2$:
 * $3 = 1 + 2$


 * The $4$th integer $n$ after $0$, $1$, $2$ such that $2^n$ contains no zero in its decimal representation:
 * $2^3 = 8$


 * The $4$th integer $n$ after $0$, $1$, $2$ such that $5^n$ contains no zero in its decimal representation:
 * $5^3 = 125$


 * The $4$th integer $n$ after $0$, $1$, $2$ such that both $2^n$ and $5^n$ have no zeroes:
 * $2^3 = 8$, $5^3 = 125$


 * The $4$th integer after $0$, $1$, $2$ which is (trivially) the sum of the increasing powers of its digits taken in order:
 * $3^1 = 3$


 * The $4$th integer $m$ after $0$, $1$, $2$ such that $m! + 1$ (its factorial plus $1$) is prime:
 * $3! + 1 = 6 + 1 = 7$


 * The $4$th palindromic integer after $0$, $1$, $2$ which is the index of a palindromic triangular number
 * $T_3 = 6$


 * The $4$th palindromic integer after $0$, $1$, $2$ whose square is also palindromic integer
 * $3^2 = 9$

Also see

 * Trisecting the Angle
 * Three Points Describe a Circle
 * Three Regular Tessellations
 * Integer is Sum of Three Triangular Numbers
 * Integer as Sum of Three Squares
 * Divisibility by 3
 * Smallest Magic Square is of Order 3
 * Prime Fibonacci Number has Prime Index except for 3

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