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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}
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  • $\begingroup$ If $\omega(2024+k)\ne 1$ , we have $f(k)=0$. $\endgroup$
    – Peter
    Commented May 20 at 4:15
  • $\begingroup$ I might include the first few values of $f$ in your Question between its definition and the summary of the scanning through its values searching for first occurrences. $\endgroup$
    – Eric Towers
    Commented May 20 at 4:25
  • $\begingroup$ To get $f(k)>7$ , we must have $k>10^{10}$ $\endgroup$
    – Peter
    Commented May 20 at 7:59
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    $\begingroup$ What have you tried? $\endgroup$
    – Hayatsu
    Commented Jun 6 at 2:45
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    $\begingroup$ @Hayatsu Is this a joke ? $\endgroup$
    – Peter
    Commented Jun 6 at 5:55

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