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Posting in MO since this questions has been unanswered in MSE for 3 months. Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $ …

Note: This question was posted in MSE about two years ago but it not receive an answer. Hence posting in MO. A positive integer is said to have square-sum divisors if it has at least two divisors whos …

Note: Posted in MO since it was unanswered in MSE. I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the …

Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it …

Consider the nine consecutive natural numbers starting from $1584614377$. n = 1584614377 no. of divisors: 2 n = 1584614378 no. of divisors: 4 n = 1584614379 no. of divisors: 8 n = 1584614380 no. of di …

Is the following true? Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$ $$ \beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a …

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a genera …