# 3003

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## Number

$3003$ (three thousand and three) is:

$3 \times 7 \times 11 \times 13$

The only known integer to appear as many as $8$ times in Pascal's triangle:
$3003 = \dbinom {3003} 1 = \dbinom {78} 2 = \dbinom {15} 5 = \dbinom {14} 6 = \dbinom {14} 8 = \dbinom {15} {10} = \dbinom {78} {76} = \dbinom {3003} {3002}$

The $6$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$ whose index is itself palindromic:
$3003 = T_{77}$

The $10$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$.

The $39$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$:
$3003 = \ds \sum_{k \mathop = 1}^{39} \paren {4 k - 3} = 39 \paren {2 \times 39 - 1}$

The $77$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2211$, $2278$, $2346$, $2415$, $2485$, $2556$, $2628$, $2701$, $2775$, $2850$, $2926$:
$3003 = \ds \sum_{k \mathop = 1}^{77} k = \dfrac {77 \times \paren {77 + 1} } 2$