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For each n, let $a_n$ be the least integer, greater than n, such that the numbers $a_n$, $a_n$+ 1, $a_n$+ 2, ..., $a_n$+ (n – 1) are divisible, in some order, by 1, 2, 3, ..., n. For example $a_{12}$ …

Murthy numbers, in a given base, are positive integers, such as 2009 in base 10, which are not relatively prime to their reversal, that is, the number written backwards (in base 10 such numbers are AO …

For each positive integer n, partition the integers 1, 2, 3, … 2n into two sets of n integers each. Let g(n) be the least integer such that there is such a partition in neither of whose parts there is …

For each positive integer $n$, let $A(n)$ be the least number such that the n residues resulting from dividing $A(n)$ by the first $n$ primes are all different. How accurate an estimate can one find f …

A positive integer n is said to be matchable if there is a coprime matching of its d(n) divisors with the set of the first d(n) positive integers, that is a bijection from one set to the other in whic …

Say two positive integers are "peers" if they are divisible by precisely the same set of primes, such as 12 and 18 (both divisible by 2 and 3), or 70 and 350 (both divisible by 2, 5 and 7). What are …

The five primes, 131, 157, 211, 349, 739, are neither in arithmetic or geometric progression, but are instead the sum of the five corresponding terms of an arithmetic and geometric progression. Are …

Given a positive integer N it is often possible to pair each of the integers 1, 2, 3, ..., N with a different integer between N + 1 and 2 N so that the product of each pair is one less or more than a …

Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?

Given a set of positive integers consider the graph whose vertices are those integers, two of which are joined by an edge if and only if they have a common divisor greater than 1 (i.e, they are not re …

Is it true that for every sufficiently large positive integer $n$, one can always find at most $k=\lfloor\pi(n)/2\rfloor$ integers, $a_1,a_2,a_3,a_3,\dots a_k$, between $1$ and $n$, such that each of …

In a recent video (https://www.facebook.com/188916357807416/videos/519169035700435/) Stephen Wolfram wonders whether, for every integer n>2, eventually the number of integers which are precisely the p …

Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence A130091 in OEIS. 24 such numbers a …

Let $G(n)$ be the graph whose vertices are the positive integers $1,2,3,4, \ldots, n$ two of which are joined by an edge if their sum is a square. Is the diameter of this graph 4 for all sufficiently …

Given a set of positive integers, its P-graph is the graph whose vertex set consists of those integers, two of which are joined by an edge if they have a common divisor greater than 1, that is, they a …