On the sequence $n^{H_{n}lcm(1,2,\cdots,n)}$, where $H_n=1+1/2+\cdots+1/n$ is the nth harmonic number and Prime Number Theorem

For an integer $\geq 1$, if $H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$ the nth harmonic number then $lcm(1,2,\cdots,n)\cdot H_n$ is an integer and

Definition. For an integer $n\geq 1$, the general term $a(n)=a_n$ of the sequence defined as $$a_n=n^{H_{n}lcm(1,2,\cdots,n)}$$ is an integer.

The sequence starts as

$$1,8,3^{11}=177147,4^{25},5^{137},\cdots$$

in a first search I don't find output as a Sloane sequence, it is I don't find that this sequence is in The On-Line Encyclopedia of Integer Sequences. I don't know if there is results about this sequence in the literature.

We know that for all $x>0$, the second Chebyshev's function is defined as $\psi(x)=\sum_{p^m\le x}\log p$, where the sum runs over all prime powers least that $x$, and satisfies $e^{\psi(n)}=lcm(1,2,\cdots,n)$, and an equivalence of Prime Number Theorem in this way $\psi(x)\sim x$ (see this Math Stack Exchange, Wikipedia or Apostol's book in Analytic Number Theory). Thus $$\log a_n= H_{n}\cdot lcm(1,2,\cdots,n)\cdot\log n$$ and then taking logarithms (we obtain for $n>1$) $$\log\log a_n=\log H_{n}+\log lcm(1,2,\cdots,n)+\log \log n.$$ Now we divide by $n$, we can write when $n$ tends to infnity that $$\lim_{n\to\infty}\frac{\log\log a_n}{n}=\lim_{n\to\infty}\frac{\log H_{n}+\log lcm(1,2,\cdots,n)+\log\log n}{n}.$$ Now, since $\lim_{n\to\infty}H_n/n=0=\lim_{n\to\infty}\log\log n/n$ we can use that $\log lcm(1,2,\cdots,n)=\psi(n)$ to wirte $\lim_{n\to\infty}\frac{\log\log a_n}{n}=0+\lim_{n\to\infty}\psi(n)/n+0$, and using Prime Number Theorem to state the following

Propostion. Let $a_n=n^{H_{n}lcm(1,2,\cdots,n)}$ then $$\log\log a_n\sim n.$$

My question is,

Question. a) Can you say more about the behaviour of $a_n$? It is you can made best computations and say more about a error term, bounds...if this question is in literature please reference to us. b) Of course, a proof verification of previous computations is required.

Thanks in advance. My only goal is edit the best post in this Mathematics Stack Exchange and learn from your computations. The following appendix isn't required to give an answer to the questions (suggestions, or contributions is not requiered but will be appreciate in comments).

Appendix (Where is exposed the context in term of thougths from where is going previous question.)

Now I tell to you my thougths, about integers $a_n$ (for integers $b_n=n^{H_{n}\cdot n!}$ I don't made computations: the procces could be the same that I tell using Stirling equivalence, this equivalence as you know is going from an equivalence that satisfies Gamma function, a function that involves the functional equation for Zeta function), are know two equivalences (elemntary) with Riemann Hypothesis

(RH1) Riemann hypothesis is equivalent with the assertion: $|L(n)-n|<(\sqrt{n})\cdot \log^2(n)$, for every integer $n\geq 3$, where $L(n)=\log lcm(1,2,\cdots,n)$ see for example [1], page 61.

(RH2) Riemann hypothesis is equivalent with the assertion: $\sigma(n)\leq H_n+e^{H_n}\log H_n$ for every integer $n\geq 1$, where $\sigma(n)=\sum_{d|n}d$ is the sum of positive divisors function. See [2].

Then I can leave the following inequality (after I explain the purpose), if you want think about this: from second derived computation in the proof of Proposition,

$$\log\log a_n=\log H_{n}+\log lcm(1,2,\cdots,n)+\log \log n$$

we mutiply by $e^{H_n}$ and add $H_n$ to write if it isn't wrong

$$H_n+e^{H_n}\log\log a_n=(H_n+e^{H_n}\log H_{n})+e^{H_n}\log lcm(1,2,\cdots,n)+e^{H_n}\log \log n.$$

Now we are looking a relation with the definition $L(n)$, $\sigma(n)$ and RH, thus we use (RH2) to write

$$H_n+e^{H_n}\log\log a_n\geq \sigma(n)+e^{H_n}\log lcm(1,2,\cdots,n)+e^{H_n}\log \log n.$$

I would like if you want now use (RH1) and your knowledge to try extract more information.

The purpose is embroil, implicate quantities and several times Riemann Hypothesis and your knowledge, in an attempt to derive statements about this unsolved problems or (conditional) relations involving cited quantities.

References:

[1] Crandall and Pomerance, Prime Numbers. A Computational Perspective, Springer 2005.

[2] Lagarias, An Elementary Problem Equivalent with Riemann Hypothesis, The American Mathematical Monthly.

• I'm sorry because I am edit several times, there was some missprints – user243301 Oct 5 '15 at 12:28
• Instead of estimating $\log\log a_n$, why not just use the equally easy-to-show $\log a_n \sim n (\log^2 n)$, which is much more fine-grained. There's nothing to this question that can't be deduced from the best known asymptotics for $\psi(n)$ and the asymptotic expansion of $H_n$. – Erick Wong Oct 5 '15 at 20:55
• I've not used your estimating since I don't known this. If you are more explicit, then I could learn from you knowledge. I'm not a professional, I know that my attempts are humbles, clumsy, perhaps vain, but I ask directly. You can edit an answer, I am waiting. Ver y thanks much @ErickWong – user243301 Oct 6 '15 at 5:55
• Of course if there is an improper part in some of my post, then if someone want give reasons, there is no reason why I want to repeat ideas. Thanks – user243301 Oct 6 '15 at 6:07
• Sorry, I think I misread your definition of $a_n$. Yeah, $\log a_n$ behaves far too erratically to get any asymptotic estimate. The reason is that when you increase $n$, sometimes $\text{lcm}(1,\ldots,n)$ jumps up by a factor of $n$, and sometimes it doesn't change at all. No simply-defined smooth function is going to interpolate that. I'm at a loss to imagine why I would want to study this particular sequence — can you? Why not just study the exponent? What's interesting about raising $n$ to the power of that integer? – Erick Wong Oct 6 '15 at 6:13