# 38

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## Number

$38$ (**thirty-eight**) is:

- $2 \times 19$

- The magic constant of the order 3 magic hexagon.

- The common sum of the $1$st triplet of consecutive positive even integers $n$ with the property $n + \map \tau n = m$ for some $m$:
- $38 = 30 + \map \tau {30} = 32 + \map \tau {32} = 34 + \map \tau {34}$

- The $1$st positive integer whose square ends in $444$:
- $38^2 = 1444$

- The $3$rd after $4$, $13$ in the sequence formed by adding the squares of the first $n$ primes:
- $38 = \displaystyle \sum_{i \mathop = 1}^3 {p_i}^2 = 2^2 + 3^2 + 5^2$

- The $4$th nontotient after $14$, $26$, $34$:
- $\nexists m \in \Z_{>0}: \map \phi m = 38$

- where $\map \phi m$ denotes the Euler $\phi$ function

- The $5$th integer after $7$, $13$, $19$, $35$ the decimal representation of whose square can be split into two parts which are each themselves square:
- $38^2 = 1444$; $144 = 12^2$, $4 = 2^2$

- The $10$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
- $3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$

- The $14$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$:
- $38 = 2 \times 19$

- The $14$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$:
- $36 = 2 + 36$

- The $14$th and largest even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$, $26$, $28$, $32$ which cannot be expressed as the sum of $2$ composite odd numbers.

- The $15$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $\ldots$

- The $24$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$, $37$ which cannot be expressed as the sum of distinct pentagonal numbers.

## Also see

- Magic Constant of Order 3 Magic Hexagon
- Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal

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## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $38$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $38$