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I did not find the following sequence in OEIS :

Number of Carmichael numbers with greatest prime factor $\le p_n$, where $p_n$ denotes the $n-th$ prime.

I want to approve my PARI/GP calculation : There are $90$ Carmichael-numbers with largest prime factor $\le 113=p_{30}$. So, $a(30)=90$.

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  • $\begingroup$ $a(29)$ should be $70$. $\endgroup$
    – Peter
    Oct 20, 2015 at 19:21
  • $\begingroup$ Since the topic is obviously of such great importance to you, take six hours of your time to sift through the $370$ OEIS entries containing the word Carmichael in the body of their post. This way you'll become better acquainted with the many relevant results in the field. Who knows ? Maybe the answer to your many questions hides in places you haven't even thought of looking in the first place. $\endgroup$
    – Lucian
    Oct 20, 2015 at 23:33

2 Answers 2

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Such a sequence would contain

2, 3, 3, 4, 6

which is the number of Carmichael numbers with greatest prime factor <= 17, 19, 23, 29, and 31 respectively. (This arises from a simple count from the first 10,000 Carmichael numbers, noting that $3\cdot5\cdot7\cdot11\cdots31$ is less than the 10,000-th Carmichael number.) The next term would be at least 8. The sequence might have some 0s out front depending on where the sequence was defined to start. The only sequence in the OEIS matching that description is A091275 but that has 9, 9 following the 6 while there are at least 10 Carmichael numbers with no prime factors greater than 41, to wit: $$\{561, 1105, 1729, 2465, 2821, 6601, 41041, 63973, 75361, 1050985\}$$

So the sequence is not in the OEIS.

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I don't see it in there.

The closest I could find was A081702, "Largest prime factor of the n-th Carmichael number." If you can construct that sequence amply, you can construct yours easily enough, but again, I don't see yours explicitly.

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