Let $\omega(n)$ be the number of distinct prime factors of the positive integer $n$.
For a positive integer $k$ , let $s$ be the smallest positive integer such that $\omega(2024^s+k)\ne s$ , in other words $\omega(2024^j+k)=j$ for every positive integer $j<s$. Then , define $f(k):=s-1$
In other words , we search numbers $k$ such that we have $\omega(2024^j+k)=j$ for $j=1,2,\cdots,t$ , but not for $t+1$. In this case , we have $f(k)=t$.
The smallest positive integers $k$ with $f(k)=1,2,3,4,5,6,7$ respective are listed in the following table :
1 3
2 63
3 15
4 879
5 49145
6 701765
7 812484789
Is $f(k)$ surjective ? In other words , can $f(k)$ be any given positive integer $t$ ?
Who can extend the above table ? What are the smallest integers $k$ giving $f(k)=8,9,10,\cdots$ ?
Remarks :
- The function is based on the current year , but it is not from a contest.
- The sequence is currently not in OEIS
- The function for PARI/GP is f(k)={j=1;while(omega(2024^j+k)==j,j=j+1);j-1}