While there have been many numbers that have been deemed 'lucky' or 'unlucky', 7 and 13 are two of the most prominently known.
So, this had led me to wonder if there were any connections between 7 and 13, and found the following.
$1$. $(7,13)$ is the largest integer pair $(n,m)$ for which ($n$th Fibonacci Number)=($n$th odd number)=$m$.
$2.$In the decimal expansion of $\log(7)$, which is $1.9459101490553\dots$, the $7$th digit after the decimal point is $1$, while the $13$th digit is $3$, giving us $13$.
Are there any other unique, special connections between $7$ and $13$?
By unique I mean that you cannot say something like $7$ and $13$ are both prime numbers or something of the sort.
Any other connections would be greatly appreciated if provided.
You may found as many as you want. A little list
(1) There are $\mathbf{7}$ nondegenerate triangles you can make with sides in $\{1,2,3\}$ and $\mathbf{13}$ you can make with sides in $\{1,2,3,4\}$.
(2) Between $2^5$ and $2^6$ there are $\mathbf{7}$ primes, and in the next such interval from $2^6$ to $2^7$ there are $\mathbf{13}$ primes.
(3) They are the two smallest numbers which are not the difference between two primes.
(4) The two smallest primes $p$ such that the next prime is $p+4$.
(5) The two smallest primes $p$ such that $2p+1$ is composite.
(6) There are $\mathbf{7}$ bipartite graphs with 4 vertices and $\mathbf{13}$ bipartite graphs with 5 vertices.
(7) They are the two smallest $n$ such that $(\mathbf{7}^n+8^n)/(\mathbf{7}+8)$ is prime. Also true for $(\mathbf{13}^n+14^n)/(\mathbf{13}+14)$.
(8) The two smallest numbers $n$ such that $(n^2 + 1)/10$ is prime.
(9) The two smallest integers $n$ such that $\sin(n)<\cos(n)<\sin(n+1)$ (in radiants).
(10) The two smallest numbers $n$ such that the decimal part of $\sqrt{n}$ starts with digit $6=\mathbf{13}-\mathbf{7}$.
(11) The two smallest primes $p$ such that neither $x^3\equiv 2\pmod p$ nor $x^3\equiv 3\pmod p$ has a solution.
(12) The two smallest numbers $n$ such that $n^2$ (here 49 and 169) is a concatenation of two nonzero squares.
(13) The two smallest numbers such that $3^n$ ends with the same digit as $n$. Here $3^7=2187$ and $3^{13}=1594323$.
(13') $7=\lfloor e^2\rfloor$, and $13=\lfloor(e+1)^2\rfloor$.
(13") The sum of digits of $7^7\cdot 13^{13}$ and the sum of digits of $7^{13}\cdot 13^7$ are both equal to $7\cdot 13$. Probably they are the only two distinct numbers with this property.