query: Each digit of $\dfrac{n(n+1)}{2}$ equals $a$
positive_passages.docid: 136885
positive_passages.text: Series of natural numbers which has all same digits
negative_passages.docid: 115145
negative_passages.text: Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits
negative_passages.docid: 144890
negative_passages.text: Recurrence $f_{n+2}=af_{n+1}+bf_n$
negative_passages.docid: 45456
negative_passages.text: Do the digits of $\pi$ contain every possible finite-length digit sequence?
negative_passages.docid: 92966
negative_passages.text: All integer values of $\frac{a^2+b^2+1}{ab-1}$
negative_passages.docid: 19588
negative_passages.text: $n!+1$ being a perfect square
negative_passages.docid: 39915
negative_passages.text: Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number
negative_passages.docid: 52157
negative_passages.text: Prove that $d(n)\leq 2\sqrt{n}$
negative_passages.docid: 72600
negative_passages.text: Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
negative_passages.docid: 106611
negative_passages.text: Prove you can weigh any number between 1 and $\frac{3^{n+1} -1}{2}$ using $n+1$ weights - Discrete
negative_passages.docid: 139671
negative_passages.text: Probability that the first digit of $2^{n}$ is 1
negative_passages.docid: 33495
negative_passages.text: A limit problem about $a_{n+1}=a_n+\frac{n}{a_n}$
negative_passages.docid: 43940
negative_passages.text: Showing that $n$ is pseudoprime to the base $a$
negative_passages.docid: 91105
negative_passages.text: For every $n\ge3$ there exists $n$ different integers $a_1,...,a_n$ so that $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$
negative_passages.docid: 77547
negative_passages.text: Find integers $a,b,c,d$ such that
negative_passages.docid: 59479
negative_passages.text: For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
negative_passages.docid: 146546
negative_passages.text: Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$