Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$ (e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it follows that $$s_b(n) > \left(\tfrac{0+1+\cdots + (b-1)}{b}-\varepsilon\right) \log_b n$$ for almost all the positive integers $n \leq x$, with at most $o(x)$ exceptions, as $x \to +\infty$. Therefore, if $\{a_k\}_{k=1}^\infty$ is a "generic" monotone increasing sequence of positive integer, it is quite reasonable to expect that $s_b(a_k) > C \log(a_k)$ for all $k$, where $C$ is a positive constant (depending on the sequence $\{a_k\}_{k=1}^\infty$). By "generic" I means, informally speaking, that $\{a_k\}_{k=1}^\infty$ has not some trivial properties that makes the statement false (e.g., $a_k := b^k$). So for example, we can conjecture that $$s_b(n!) > C_1 n \log n$$ $$s_b(a^n) > C_2 n$$ $$s_b(F_n) > C_3 n$$ for all sufficiently large $n$, where $a$ is a positive integer coprime to $b$ and $F_n$ is the $n$-th Fibonacci number. In [1] and [2] they have been shown the weaker lower bounds $s_b(n!) > C_1 \log n$ and $s_b(F_n) > C_2 \log n / \log \log n$, respectively.

My question is: Are there some non(too)trivial example of $\{a_k\}_{k=1}^\infty$ such that the optimal lower bound $s_b(a_k) > C \log(a_k)$ has been proved?

Thank you very much for any suggestion/reference.

[1] Luca F., The number of nonzero digits of n!, Canad. Math. Bull., 45, 2002, 115–11

[2] Luca F., Distinct digits in base b expansions of linear recurrence sequences, Quaest. Math., 23, 2000, 389–404.

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$ (e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it follows that $$s_b(n) > \left(\tfrac{0+1+\cdots + (b-1)}{b}-\varepsilon\right) \log_b n$$ for almost all the positive integers $n \leq x$, with at most $o(x)$ exceptions, as $x \to +\infty$. Therefore, if $\{a_k\}_{k=1}^\infty$ is a "generic" monotone increasing sequence of positive integer, it is quite reasonable to expect that $s_b(a_k) > C \log(a_k)$ for almost all $k$, i.e., with the exception of a set of asymptotic density 0, where $C$ is a positive constant (depending on the sequence $\{a_k\}_{k=1}^\infty$). By "generic" I means, informally speaking, that $\{a_k\}_{k=1}^\infty$ has not some trivial properties that makes the statement false (e.g., $a_k := b^k$). So for example, we can conjecture that $$s_b(n!) > C_1 n \log n$$ $$s_b(a^n) > C_2 n$$ $$s_b(F_n) > C_3 n$$ for almost all $n$, where $a$ is a positive integer coprime to $b$ and $F_n$ is the $n$-th Fibonacci number. In [1] and [3] they have been shown the weaker lower bounds $s_b(n!) > C_1 \log n \log \log \log n$ (see also [2]) and $s_b(F_n) > C_2 \log n / \log \log n$, for all integers $n$, respectively. Moreover, in [4] the bound $s_b(a_n) > C \log n$ for almost all $n$, has been proved for any sequence of integers $$a_n = e^{f(n)} (1 + O(n^{-\alpha}))$$ where $\alpha > 0$ and $f(x)$ is a two times differentiable function satisfying $f(x) \asymp 1/x$ for large x.

My question is: Are there some non(too)trivial example of $\{a_k\}_{k=1}^\infty$ such that the optimal lower bound $s_b(a_k) > C \log(a_k)$, for almost all $k$, has been proved?

Thank you very much for any suggestion/reference.

[1] C. Sanna, On the sum of digits of the factorial, J. Number Theory 147, 2015, 836--841.

[2] F. Luca, The number of nonzero digits of n!, Canad. Math. Bull., 45, 2002, 115--11.

[3] F. Luca, Distinct digits in base b expansions of linear recurrence sequences, Quaest. Math., 23, 2000, 389--404.

[4] J. Cilleruelo, F. Luca, J. Rué and A. Zumalacárregui, On the sums of digits of the some sequences of integers, Cent. Eur. J. Math., 11, 2013, 188--195.

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$ (e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it follows that $s_b(n) > (1+2+\cdots+(b-1)) \log_b n$ for almost all the positive integers $n \leq x$, with at most $o(x)$ exceptions, as $x \to +\infty$. Therefore, if $\{a_k\}_{k=1}^\infty$ is a "generic" monotone increasing sequence of positive integer, it is quite reasonable to expect that $s_b(a_k) > C \log(a_k)$ for all $k$, where $C$ is a positive constant (depending on the sequence $\{a_k\}_{k=1}^\infty$). By "generic" I means, informally speaking, that $\{a_k\}_{k=1}^\infty$ has not some trivial properties that makes the statement false (e.g., $a_k := b^k$). So for example, we can conjecture that $$s_b(n!) > C_1 n \log n$$ $$s_b(a^n) > C_2 n$$ $$s_b(F_n) > C_3 n$$ for all sufficiently large $n$, where $a$ is a positive integer coprime to $b$ and $F_n$ is the $n$-th Fibonacci number. In [1] and [2] they have been shown the weaker lower bounds $s_b(n!) > C_1 \log n$ and $s_b(F_n) > C_2 \log n / \log \log n$, respectively.

My question is: Are there some non(too)trivial example of $\{a_k\}_{k=1}^\infty$ such that the optimal lower bound $s_b(a_k) > C \log(a_k)$ has been proved?

Thank you very much for any suggestion/reference.

[1] Luca F., The number of nonzero digits of n!, Canad. Math. Bull., 45, 2002, 115–11

[2] Luca F., Distinct digits in base b expansions of linear recurrence sequences, Quaest. Math., 23, 2000, 389–404.

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$ (e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it follows that $$s_b(n) > \left(\tfrac{0+1+\cdots + (b-1)}{b}-\varepsilon\right) \log_b n$$ for almost all the positive integers $n \leq x$, with at most $o(x)$ exceptions, as $x \to +\infty$. Therefore, if $\{a_k\}_{k=1}^\infty$ is a "generic" monotone increasing sequence of positive integer, it is quite reasonable to expect that $s_b(a_k) > C \log(a_k)$ for all $k$, where $C$ is a positive constant (depending on the sequence $\{a_k\}_{k=1}^\infty$). By "generic" I means, informally speaking, that $\{a_k\}_{k=1}^\infty$ has not some trivial properties that makes the statement false (e.g., $a_k := b^k$). So for example, we can conjecture that $$s_b(n!) > C_1 n \log n$$ $$s_b(a^n) > C_2 n$$ $$s_b(F_n) > C_3 n$$ for all sufficiently large $n$, where $a$ is a positive integer coprime to $b$ and $F_n$ is the $n$-th Fibonacci number. In [1] and [2] they have been shown the weaker lower bounds $s_b(n!) > C_1 \log n$ and $s_b(F_n) > C_2 \log n / \log \log n$, respectively.

My question is: Are there some non(too)trivial example of $\{a_k\}_{k=1}^\infty$ such that the optimal lower bound $s_b(a_k) > C \log(a_k)$ has been proved?

Thank you very much for any suggestion/reference.

[1] Luca F., The number of nonzero digits of n!, Canad. Math. Bull., 45, 2002, 115–11

[2] Luca F., Distinct digits in base b expansions of linear recurrence sequences, Quaest. Math., 23, 2000, 389–404.