# 49

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## Number

$49$ (**forty-nine**) is:

- $7^2$

- The $1$st square whose decimal representation can be split into two parts which are each themselves square:
- $49 = 7^2$; $4 = 2^2$, $9 = 3^2$

- The $1$st term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
- $\tuple {49, 50, 51, 52, 53}$

- The $2$nd power of $7$ after $(1)$, $7$:
- $49 = 7^2$

- The $4$th square lucky number:
- $1$, $9$, $25$, $49$, $\ldots$

- The $7$th square number after $1$, $4$, $9$, $16$, $25$, $36$:
- $49 = 7 \times 7$

- The $7$th square after $1$, $4$, $9$, $16$, $25$, $36$ which has no more than $2$ distinct digits and does not end in $0$:
- $49 = 7^2$

- The $7$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$ such that the Fibonacci number $F_n$ ends in $n$

- The $8$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$:
- $49^3 = 117 \, 6 \mathbf {49}$

- The $10$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$

- The $11$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$:
- $49 \to 4^2 + 9^2 = 16 + 81 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$

- The $13$th lucky number:
- $1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $\ldots$

- The $17$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$, $46$:
- $49 = 7 \times 7$

- The $24$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
- $1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $\ldots$

- The $29$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
- $2^{49} = 562 \, 949 \, 953 \, 421 \, 312$

- The $30$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$, $45$, $46$ which cannot be expressed as the sum of distinct pentagonal numbers.

- The $33$rd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

## Also see

*Previous ... Next*: Square Number*Previous ... Next*: Powerful Number*Previous ... Next*: Squares with No More than 2 Distinct Digits

*Previous ... Next*: Lucky Number

*Previous ... Next*: Happy Number

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Semiprime Number

*Next*: Quintuplets of Consecutive Integers which are not Divisor Sum Values*Next*: Squares whose Digits can be Separated into 2 other Squares

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $49$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $49$