Divisors and Multiples

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E0018/: Asks how few distinct divisors of a practical number represent every smaller integer, in particular for factorials; h(n!) < n^{o(1)} is proved (Conjectures.io, 2026), the (log log m)^{O(1)} part claimed, the rest open.

E0026/: Asks whether every infinite set of natural numbers has a shift k for which almost all integers have a divisor that is a member plus k.

E0056/: Asks whether the multiples of the first k primes form the largest subset of the first N integers with no k plus one pairwise relatively prime elements.

E0143/: Asks whether a countable set of reals above one where every integer multiple of an element is at distance one or more from another must be sparse.

E0144/: Asks whether almost every integer has two divisors with the larger less than twice the smaller, so that the density of such integers exists and equals one.

E0164/: Asks whether the sum of one over n times the logarithm of n over a set with no member dividing another is largest when the set is the primes.

E0381/: Asks whether the number of highly composite numbers up to x grows faster than any fixed power of the logarithm of x.

E0444/: Asks whether, for every k, an infinite set of integers has some integer below x with more divisors from the set than any power of the set's reciprocal sum.

E0446/: The growth rate of the density of integers having a divisor strictly between n and twice n.

E0448/: Asks whether, for almost all n, the number of dyadic ranges containing a divisor of n is an arbitrarily small fraction of the total number of divisors.

E0449/: Asks whether, for almost all n, the number of pairs of divisors within a factor of two of each other is an arbitrarily small fraction of the divisor count.

E0450/: How long an interval must be so that at most a small fraction of its integers have a divisor strictly between n and twice n.

E0468/: Studies the set of partial sums of the increasing divisors of n above one.

E0469/: Asks whether the reciprocal sum converges over integers that are sums of distinct proper divisors of themselves while no proper divisor has that property.

E0470/: Studies weird numbers, those whose divisor sum is at least twice the number yet which are not the sum of any set of their own divisors.

E0486/: Asks whether the set of integers avoiding a prescribed residue class pattern modulo each member of a given set of moduli always has a logarithmic density.

E0534/: The largest subset of the integers up to N that contains N itself and in which every two distinct elements share a common factor greater than one.

E0673/: Asks whether the sum of the ratios of consecutive divisors of n tends to infinity for almost all n, and seeks an asymptotic formula for its average.

E0692/: Asks whether the density of integers with exactly one divisor in an interval from n to m is unimodular in m; disproved by Cambie, who also shows many local maxima.

E0693/: Bounds the largest gap between consecutive integers above n having a divisor between n and twice n, asking if it is at most a power of the logarithm of n.

E0696/: The longest chain of primes dividing n, and the longest chain of divisors of n, in which each term is congruent to 1 modulo the previous term.

E0697/: Asks whether some threshold exponent splits the density of integers with a divisor above 1 congruent to 1 modulo m into a limit of zero below and one above.

E0844/: Bounds the largest set of integers up to N in which the product of any two members is never squarefree.

E0858/: Estimates the largest reciprocal sum, over the logarithm of N, of a set of integers up to N in which no member equals another member times a factor whose prime factors all exceed the smaller member.

E0859/: Asks whether the density of the integers n for which a given t is a sum of distinct divisors of n is asymptotic to a constant over a power of log t; false by a Lean disproof the bounty site Conjectures.io certified in 2026.

E0872/: How long the game in which two players alternately add integers up to n to a shared set free of divisibility between members can be guaranteed to last, and whether it lasts at least εn or (1-ε)n/2 moves.

E0882/: The size of the largest subset of one to n whose nonempty subset sums form a set in which no element divides another.

E0884/: Asks whether the sum of one over all differences of divisors of n is bounded by a constant times one plus the sum of one over consecutive divisor gaps.

E0885/: Asks whether, for every k, there are k integers whose sets of differences of complementary factor pairs share at least k common values.

E0886/: Asks whether, for each fixed positive epsilon, every large n has only boundedly many divisors just above the square root of n.

E0887/: Asks whether there is an absolute bound on the number of divisors of a large n lying within a constant times the fourth root of n above its square root.

E0892/: Asks for a necessary and sufficient condition on an increasing sequence for a primitive sequence, no term dividing another, to grow no faster than it.

E0893/: Asks whether the ratio of the summed divisor counts of two to the power k minus one, over k up to twice n and up to n, tends to a limit.

E0945/: Estimates the longest run of consecutive integers below x with all divisor counts distinct, and whether short intervals must repeat a divisor count.

E0946/: Asks whether there are infinitely many n for which n and n plus 1 have the same number of divisors.

E0964/: Asks whether the ratios of the number of divisors of n plus one to the number of divisors of n are dense in the positive reals.

E1054/: Studies the least m such that n is the sum of the k smallest divisors of m for some k, and how that function behaves.

E1099/: Asks whether the sum of the ratios of consecutive divisors of n minus one, each raised to a power alpha above one, has bounded limit inferior over all n.

E1100/: Concerns the number of consecutive pairs of divisors of n that are coprime.

E1196/: Asks whether the sum of one over a times log a over a primitive set of integers all at least x is at most one plus a quantity tending to zero.

E1217/: Asks whether a sequence of positive lower logarithmic density contains a divisibility chain whose upper growth rate against log log x is at least the weighted sum's; answered yes in 2026 by Alexeev and seven coauthors.

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Divisor functions and the distribution of divisors, multiplication tables, sets of multiples, primitive sets in which no element divides another, and other sets of integers defined by divisibility or coprimality conditions.

Site tags routed here: divisors, factorials, intersecting family, number theory, primitive sets.