35
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Number
$35$ (thirty-five) is:
- $5 \times 7$
- The $1$st positive integer $n$ such that $\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$:
- $\map {\sigma_1} {35} = 48 = \dfrac {\map \phi {35} \times \map {\sigma_0} {35} } 2$
- The $2$nd positive integer after $1$ whose square can be expressed as the sum of a sequence of odd cubes from $1$:
- $35^2 = 1225 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3$
- The number of pairs of twin primes less than $10^3$
- The $4$th pentatope number after $1$, $5$, $15$:
- $35 = 1 + 4 + 10 + 20 = \dfrac {4 \paren {4 + 1} \paren {4 + 2} \paren {4 + 3} } {24}$
- The $4$th integer after $7$, $13$, $19$ the decimal representation of whose square can be split into two parts which are each themselves square:
- $35^2 = 1225$; $1 = 1^2$, $225 = 15^2$
- The $5$th pentagonal number after $1$, $5$, $12$, $22$:
- $35 = 1 + 4 + 7 + 10 + 13 = \dfrac {5 \paren {3 \times 5 - 1} } 2$
- The $5$th tetrahedral number, after $1$, $4$, $10$, $20$:
- $35 = 1 + 3 + 6 + 10 + 15 = \dfrac {5 \paren {5 + 1} \paren {5 + 2} } 6$
- The $5$th integer $n$ after $1$, $3$, $15$, $30$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
- $\map {\sigma_0} {35} = 4$, $\map \phi {35} = 24$, $\map {\sigma_1} {35} = 48$
- The number of distinct free hexominoes
- The $9$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$:
- $35 = \dfrac {5 \paren {3 \times 5 - 1} } 2$
- The $13$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$:
- $35 = 5 \times 7$
- The $18$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
- $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $35$, $\ldots$
- The $19$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes
- The $25$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$, $26$, $27$, $30$, $31$, $32$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
- The $25$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
- $2^{35} = 34 \, 359 \, 738 \, 368$
- The maximum length of a non-crossing knight's tour on a standard chessboard.
- $35$ and $4374$ have the same prime factors between them as $36$ and $4375$:
- $35 = 5 \times 7$, $4374 = 2 \times 3^7$; $36 = 2^2 \times 3^2$, $4375 = 5^4 \times 7$
Arithmetic Functions on $35$
\(\ds \map {\sigma_0} { 35 }\) | \(=\) | \(\ds 4\) | $\sigma_0$ of $35$ | |||||||||||
\(\ds \map \phi { 35 }\) | \(=\) | \(\ds 24\) | $\phi$ of $35$ | |||||||||||
\(\ds \map {\sigma_1} { 35 }\) | \(=\) | \(\ds 48\) | $\sigma_1$ of $35$ |
Also see
- Previous ... Next: Pentatope Number
- Previous ... Next: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes
- Previous ... Next: Integers not Expressible as Sum of Distinct Primes of form 6n-1
- Previous ... Next: Powers of 2 with no Zero in Decimal Representation
- Previous ... Next: Semiprime Number
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $35$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$