Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum_{n=1}^\infty \Big(1-\zeta(n)^{-1}\Big)$$ with the convention $\zeta(1)^{-1} = 0$ (for aesthetics). I was just wondering whether this constant $\eta$ has a name, whether it's been studied, etc..
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$\begingroup$ What do you mean by “random positive integer”? $\endgroup$– Alessandro Della CorteMay 12 at 9:35
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1$\begingroup$ @AlessandroDellaCorte I guess my last comment provided the definition for "largest exponent in prime factorization of a random positive integer" rather than for "a random positive integer" which is undefined. $\endgroup$– mathworker21May 12 at 10:01
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2$\begingroup$ 👍(and since some more characters are needed, I add these ones) $\endgroup$– Alessandro Della CorteMay 12 at 10:59
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7$\begingroup$ Don't you mean to set $\zeta(1)^{-1}=-12$? (Joke, joke! Put down the pitchforks...) $\endgroup$– Steven StadnickiMay 12 at 16:31
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2$\begingroup$ @StevenStadnicki You're thinking of $\zeta(-1)^{-1}=-12$ (down, pitchforks, down). I may have just accidentally explained the joke. $\endgroup$– J.G.May 13 at 8:16
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1 Answer
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If you calculate it to a few decimals, you find $$ 1.705211140105\ldots $$ which is enough to locate it in the OEIS.
It's Niven's constant: MathWorld, Wikipedia, OEIS.
As mentioned by GH from MO in the comments, it was in fact proven by Niven in 1969 that the average largest exponent tends to $\eta$.
Since the question is essentially about finding literature related to a given numerical constant, I should probably mention this answer by myself on the other site, exhibiting some other methods (Steven Finch's book Mathematical Constants & how to google the decimals effectively).
answered May 12 at 11:31
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4$\begingroup$ The OP's $\eta$ is Niven's constant. The OP's term corresponding to $n=1$ equals $1$. More importantly, it seems that the OP rediscovered Niven's theorem from 1969 (doi.org/10.2307%2F2037055). $\endgroup$ May 12 at 13:53
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7$\begingroup$ Thanks! You just added a new tool to my toolbox: I've been using inverse symbolic calculators instead of inputting the digits as a sequence to the OEIS! Also, as GH said, $1$ must be added. $\endgroup$ May 12 at 15:01
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7$\begingroup$ Glad if it helps! BTW you can input it into OEIS either as a sequence of digits ("7,0,5,2,1,1,1") or more conveniently as the decimal (".7052111"). I didn't know the latter one works until I tried it today. $\endgroup$ May 12 at 15:26
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1$\begingroup$ GH, thanks, somehow I misread the first term as zero. Will fix that in my answer. $\endgroup$ May 12 at 15:27
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7$\begingroup$ @JukkaKohonen It would be weird if the average of the largest exponent were less than $1$ :) $\endgroup$ May 12 at 17:15