210

Number
$210$ (two hundred and ten) is:


 * $2 \times 3 \times 5 \times 7$


 * The $2$nd pentagonal number after $1$ which is also triangular:
 * $210 = \dfrac {12 \paren {3 \times 12 - 1} } 2 = \dfrac {20 \times \paren {20 + 1} } 2$


 * The $3$rd triangular number after $6$, $120$ which can be expressed as the product of $3$ consecutive integers:
 * $210 = T_{20} = 5 \times 6 \times 7$


 * The $4$th primorial after $1$, $2$, $6$, $30$ (counting $1$ as the zeroth):
 * $210 = p_4 \# = 7 \# = 2 \times 3 \times 5 \times 7$
 * Hence the smallest positive integer with $4$ distinct prime factors.


 * The $4$th primorial after $2$, $6$, $30$ which can be expressed as the product of consecutive integers:
 * $210 = 7 \# = 14 \times 15$


 * The $7$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$:
 * $210 = 1 + 4 + 10 + 20 + 35 + 56 + 84 = \dfrac {7 \paren {7 + 1} \paren {7 + 2} \paren {7 + 3} } {24}$


 * The $11$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$ whose $\sigma$ value is square:


 * The $12$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$:
 * $210 = 1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 + 34 = \dfrac {12 \paren {3 \times 12 - 1} } 2$


 * The $12$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$.


 * The $13$th integer $n$ after $1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map \sigma n$:
 * $\map {\sigma_0} {210} = 16$, $\map \phi {210} = 48$, $\map \sigma {210} = 576$


 * The $20$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$, $91$, $105$, $120$, $136$, $153$, $171$, $190$:
 * $210 = \ds \sum_{k \mathop = 1}^{20} k = \dfrac {20 \times \paren {20 + 1} } 2$


 * The $21$st and last of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
 * $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$, $42$, $48$, $60$, $90$, $210$


 * The $23$rd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $100$, $117$, $126$, $145$, $155$, $176$, $187$:
 * $210 = \dfrac {12 \paren {3 \times 12 - 1} } 2$


 * The $27$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$, $180$:
 * $\map \sigma {210} = 576$


 * The $41$st positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.

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