The sequence $n,\phi (n),\phi (\phi (n)),\dots ,1$ obtained by iterating Euler's totient function on a positive integer. Since $1\le \phi (n)<n$ for $n>1$, the sequence reaches 1. Counting both endpoints gives the length recurrence relation $L(1)=1$ and $L(n)=1+L(\phi (n))$; increasing order of $n$ therefore suffices for dynamic programming.

For an integer $d>1$, the resilience of a denominator is the fraction of its proper fractions $1/d,2/d,\dots ,(d-1)/d$ already in lowest terms. The fraction $a/d$ is a reduced fraction exactly when $a$ and $d$ are coprime, sowhere $\phi$ is Euler's totient function.

For $d=12$, the allowed numerators are $1,5,7,11$, giving $R(12)=4/11$. A prime denominator has resilience one. Conversely, a composite number has a proper prime divisor $p$, and $p/d$ is among the listed fractions but is not reduced; thus its resilience is strictly less than one.

The prime factorization formula givesThe correction $d/(d-1)$ comes from sampling the $d-1$ positive proper numerators, rather than all $d$ residue classes. The related coresilience uses $C(d)=(d-\phi (d))/(d-1)$.

For an integer $n>1$, the coresilience iswhere $\phi$ is Euler's totient function.^[ref] It is related to the resilience of a denominator $R(n)=\phi (n)/(n-1)$, but is not its complement:For a prime number $p$, $R(p)=1$ but $C(p)=1/(p-1)$, rather than zero. For the composite number fifteen, $\phi (15)=8$ and $C(15)=7/14=1/2$.

As $n-\phi (n)>0$, the value is a unit fraction precisely when $n-\phi (n)$ divides $n-1$. A composite prime power $n=p^k$ with $k\ge 2$ cannot qualify: its value is $p^{k-1}/(p^k-1)$, already a reduced fraction with numerator greater than one. This excludes an entire family of composite numbers before a search.

The summatory totient function is $\Phi (x)={\sum }_{n\le x}\phi (n)$. It counts the coprime ordered pairs $(a,b)$ with $1\le a\le b\le x$. The identity ${\sum }_{d\mid n}\phi (d)=n$ yields ${\sum }_{j=1}^x\Phi (\lfloor x/j\rfloor )=x(x+1)/2$ for a positive integer $x$.

Given a positive integer $m$, find all positive integers $n$ satisfying $\phi (n)=m$. For every prime divisor $p$ of a solution, $p-1$ divides $m$, so the possible primes lie in the finite set $\{d+1:d\mid m,d+1\text{ is prime}\}$. Search can then combine the prime powers whose totient contributions multiply to $m$.

For a positive integer $n$, Euler's totient product formula iswith the product over distinct prime divisors. Among the residues $1,\dots ,n$, a fraction $1/p$ is divisible by $p$. Inclusion-exclusion over the prime divisors removes exactly those residues not coprime to $n$.

For $n=12$, the formula gives $12(1-1/2)(1-1/3)=4$, corresponding to $1,5,7,11$. It gives $\phi (p^a)=p^{a-1}(p-1)$ and shows that the Euler totient function is multiplicative on coprime integers. For $n=1$, the empty product gives $\phi (1)=1$.

Two intersecting straight lines are perpendicular if their angle is $90^{\circ }$. Their direction vectors have zero dot product.

The ratio between corresponding lengths of two geometrically similar figures. Multiplying every length by $t>0$ multiplies areas by $t^2$ and volumes by $t^3$. For integer-sided triangles, choosing a positive integer scale factor preserves integer side lengths.

The Sylvester-Gallai theorem states that a finite noncollinear set of points in the real Euclidean plane determines a line containing exactly two of its points. Finiteness, noncollinearity and the real field matter. A useful proof chooses the smallest positive distance between a point $P$ and a line $L$ determined by the set. If $L$ contains at least three points, two of them, $A,B$, lie on the same closed side of the perpendicular foot, with $A$ nearer. The distance from $A$ to $PB$ is the original altitude multiplied by $AB/PB<1$, contradicting minimality. Thus $L$ contains exactly two points. Related statements over complex projective space require a different conclusion.^[ref]

The dihedral angle measures the angle between two intersecting planes. Molecular torsion angles use an oriented four-atom convention to specify a related signed angle.

An etch that removes a material in different directions without a strongly preferred geometrical direction. It can undercut an overhang and release a sacrificial layer, but can also widen a patterned opening. Access and transport can make the realized undercut nonuniform even with locally isotropic chemistry.

Chemical stripping of organic material, commonly photoresist, by reactive species produced in a plasma. oxygen radicals convert suitable organic material into volatile oxidation products. A remote source reduces direct energetic ion bombardment, but exposed materials can still oxidize and inorganic residue can remain.

A durable patterned layer used to protect underlying material during an etch. Its thickness and etch selectivity must survive the intended depth; erosion and lateral bias can change final feature dimensions even when the starting pattern is correct.

An ester with a methylene group between two carbonyls. Its conjugate base is stabilized by delocalization involving both carbonyl groups.

The methyl ester of but-2-enoic acid. Its conjugated acyl group has different carbonyl and vinylic environments from an ester whose alkene lies in an allyl alcohol-derived group.

The ester CH3COOCH2CH=CH2. Its terminal vinyl hydrogens are chemically nonequivalent and have distinct cis and trans couplings to the internal vinyl proton.

An ester of a sulfonic acid. In an appropriate alkyl sulfonate, substitution at carbon can expel a resonance-stabilized sulfonate leaving group.

A tert-Butyl ester contains the group $\mathrm{R}-\mathrm{C}(=\mathrm{O})-\mathrm{OC}(\mathrm{C}{\mathrm{H}}_3{)}_3$. Its bulky alcohol-derived group differs from the smaller group in a methyl ester.

A tert-Butyl ester can protect a carboxylic acid during a synthesis and then be cleaved under suitable acidic conditions. This is an example of selecting a protecting group by the conditions required for both its survival and removal.