Question 146
An integer $n \geqslant 2$ having exactly $s$ positive divisors $1=d_{1}\lt d_{2}\lt \cdots\lt d_{s}=n$ is said to be good if there exists an integer $k$, with $2 \leqslant k \leqslant s$, such that $d_{k}\gt 1+d_{1}+\cdots+d_{k-1}$. An integer $n \geqslant 2$ is said to be bad if it is not good.
a. Show that there are infinitely many bad integers.
b. Prove that, among any seven consecutive integers all greater than $2$, there are always at least four good integers.
c. Show that there are infinitely many sequences of seven consecutive good integers.
(Gerhard Woeginger, Luxembourg)
a. Show that there are infinitely many bad integers.
b. Prove that, among any seven consecutive integers all greater than $2$, there are always at least four good integers.
c. Show that there are infinitely many sequences of seven consecutive good integers.
(Gerhard Woeginger, Luxembourg)