Sequences and Densities of Integers

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E0012/: Asks how large an infinite set with no element dividing the sum of two larger elements can be, in counting, density and reciprocal-sum terms; the first two questions have 2026 claimed answers, the reciprocal sum is open.

E0013/: Asks whether a subset of the first N integers with no element dividing the sum of two larger elements has size at most N over 3 plus a constant; proved by Bedert in 2023, with the ceiling of N over 3 exact for large N.

E0025/: Asks whether the integers avoiding a chosen residue class for each modulus in an increasing sequence always have a logarithmic density.

E0038/: Asks whether a set that is not an additive basis can still always supply a shift raising the count of any set of Schnirelmann density strictly between 0 and 1 by a positive fraction depending on the density; proved in 2026.

E0121/: Asks whether the largest subset of the first N integers with no five (or no fixed odd number, at least five, of) distinct elements multiplying to a square has size nearly N; Tao (2024) disproved it for every size at least 4.

E0131/: Estimates the largest subset of the first N integers in which no element divides the sum of any distinct others; the displayed root-N question is answered no, and the order carries one unreviewed claim of exponent 1/5.

E0145/: Asks whether the average of the alpha-th power of gaps between consecutive squarefree numbers up to x has a limit for every non-negative alpha.

E0208/: Bounds the gaps between consecutive squarefree numbers, asking whether they are smaller than any fixed power, and whether a sharp logarithmic bound holds.

E0220/: Asks whether the sum of squared gaps between consecutive integers below n and coprime to n is at most a constant times n squared over Euler's totient of n.

E0222/: Bounds the gaps between consecutive integers that are sums of two squares, from above and below.

E0235/: Asks whether the distribution of normalized gaps between integers coprime to the product of the first k primes tends to a continuous limiting function.

E0253/: Asks whether a sequence with ratios tending to one whose distinct subset sums meet every arithmetic progression infinitely often represents all large integers.

E0254/: Asks whether a set that grows in every dyadic range and has divergent sums of distances to the nearest integer for every angle represents all large integers as distinct sums.

E0329/: The largest possible value of the limiting ratio of the counting function of an infinite Sidon set to the square root of N.

E0332/: Determines which conditions on a set of integers force the differences that occur infinitely often to have bounded gaps.

E0341/: Asks whether the sequence extending a finite set by the least integer that is not a sum of two earlier terms has eventually periodic differences.

E0342/: Asks what can be said about the sequence whose terms are the least integers uniquely expressible as a sum of two earlier terms, its density and its gaps.

E0356/: Asks whether some positive c gives, for all large n, integers up to n whose sums over blocks of consecutive terms take at least c times n squared values.

E0357/: The growth rate of the largest number of integers up to n whose sums over blocks of consecutive terms are all distinct, and whether it is smaller than n.

E0359/: The density of the sequence starting at n in which each later term is the least integer that is not a sum of consecutive earlier terms.

E0360/: The growth rate of the fewest classes needed to partition the numbers below n so that n is never a sum of distinct members of one class.

E0361/: The largest subset of the integers up to c times n that has no subset summing to n, and whether its size varies irregularly with n.

E0375/: Asks whether any k consecutive composite integers can always be assigned distinct primes, one dividing each of them.

E0402/: Asks for a proof that every finite set of integers has two members whose greatest common divisor is at most one member divided by the set's size.

E0421/: Asks whether there is an increasing sequence of density one all of whose products of consecutive blocks of terms are distinct; answered yes in July 2026 by Chojecki and Sneiderman, accepted by the site, not refereed.

E0422/: Determines the behavior of a self-referential recursion whose terms are sums of earlier terms, and whether it misses infinitely many integers.

E0423/: Estimates the growth of the sequence beginning one, two in which each term is the least larger integer that is a sum of consecutive earlier terms.

E0424/: Asks whether the integers eventually produced from two and three by repeatedly adjoining products of two distinct terms minus one have positive density, read as positive lower density following the site's curator.

E0429/: Asks whether a sufficiently sparse set missing a residue class modulo every prime must have some shift all of whose members are prime; disproved by Weisenberg's 2024 arbitrarily sparse admissible sets with no prime shift.

E0430/: Concerns the greedy decreasing sequence in which each term is the largest smaller integer above 1 whose prime factors all exceed n minus that term, and asks whether some term is composite for every large n.

E0432/: Determines how dense the sumset of two infinite sets of natural numbers can be when all its elements are pairwise coprime.

E0436/: Asks whether the limiting least start of m consecutive k-th power residues modulo p is finite for m equal to two, and for m equal to three with k odd.

E0438/: The largest subset of the first N integers whose pairwise sums include no perfect square.

E0440/: Whether the number of consecutive pairs of an infinite set with least common multiple at most x is O(x^{1/2}), and how large the liminf of that count over x^{1/2} can be; both answered by Erdős and Szemerédi in 1980.

E0441/: The largest subset of one through N with all pairwise least common multiples at most N, asymptotically the square root of 9N/8 by Chen, and whether Erdős's construction attains it, which Chen and Dai refute infinitely often.

E0442/: Whether a set whose reciprocal sum grows faster than log log x forces the normalized sum of reciprocals of pairwise least common multiples to blow up; disproved by Tao, whose construction also gives the optimal growth threshold.

E0451/: Estimates the least integer above twice k for which the product of its k preceding integers has no prime factor between k and twice k; claimed in a 2026 preprint to grow faster than any power of k, and at most exponential.

E0453/: Asks whether, for all large n, some symmetric pair of primes around the nth prime has product exceeding the square of the nth prime.

E0455/: Asks whether a sequence of primes with non-decreasing gaps must have its nth term grow faster than n squared.

E0457/: Whether some epsilon > 0 gives infinitely many n with every prime up to (2 + epsilon) log n dividing the product of the next log n integers: yes, by a 2026 AI construction the site accepted; Erdős's opposite 1979 form: no.

E0460/: Asks whether the reciprocal sum of the terms below n of the greedy sequence making n minus each term pairwise coprime tends to infinity, and likewise for two subsequences.

E0467/: Asks whether, for large x, residues can be chosen for the primes up to x and those primes split in two so every integer below x is covered by both parts; open, with one sentence of the 1980 monograph as its only source.

E0471/: Asks whether some finite starting set of primes grows without bound under repeated adjunction of all primes that are sums of three distinct members; yes, by Vinogradov's three-primes theorem, an observation the site accepted.

E0472/: Asks whether some starting list of primes makes infinite the sequence whose next term is the least prime of the form previous plus an earlier term minus one.

E0473/: Asks whether the positive integers can be permuted so that every two consecutive terms sum to a prime; yes, by an unpublished construction of Odlyzko reported by Erdős and Graham in 1980 and accepted by the site.

E0481/: Asks whether iterating a family of affine maps from the value one must repeat an element when the reciprocals of the multipliers sum to over one; yes, by Klarner's 1982 theorem, its 2022 extension and a 2025 thread proof.

E0487/: Asks whether every set of integers of positive density contains three distinct members one of which is the least common multiple of the other two; true by Kleitman's union-free theorem, attested here second-hand.

E0488/: Asks whether the density of the multiples of a finite set up to m is under twice its density up to a smaller n at least the largest element; a counterexample claim of September 2026 is pending and unreviewed.

E0489/: Asks whether the average of the squared gaps between consecutive integers divisible by no member of a sparse set of divisors tends to a finite limit.

E0490/: Asks whether two subsets of the first N integers with all pairwise products distinct must have size product at most about N squared over the logarithm of N; proved by Szemerédi (1976), with a second proof by Erdős and Szemerédi.

E0535/: Estimates the largest subset of the integers up to N containing no r elements whose pairwise greatest common divisors are all equal, for r at least three.

E0536/: The largest subset of the integers up to N containing no three distinct elements whose three pairwise least common multiples are all equal.

E0537/: Asks whether every subset of the integers up to N of positive density contains three elements which become equal after multiplying each by a distinct prime.

E0538/: The best upper bound for the reciprocal sum of a set of integers up to N in which every number has at most r representations as a prime times a set element.

E0539/: Estimates the least possible size of the set of ratios of each element to the greatest common divisor of a pair, over all sets of n natural numbers.

E0540/: Asks whether any subset of the integers modulo N of size at least a constant times the square root of N has a non-empty subset summing to zero modulo N; proved by Szemerédi in 1970 for all finite abelian groups.

E0541/: Asks whether p residues modulo p, whose zero-sum non-empty subsets all have the same size, must take at most two distinct values; Graham's conjecture, proved for large primes in 1976 and for every modulus in 2010.

E0542/: Asks whether a set of integers up to n with pairwise least common multiples above n has reciprocal sum at most 31/30, and whether a positive proportion of the integers up to n avoid its multiples; yes and no, Schinzel-Szekeres.

E0587/: Determines the largest set of integers up to N in which no non-empty subset has a square sum.

E0635/: Asks how large a subset of the first N integers can be if no difference of at least t between two of its elements divides the larger element.

E0650/: Asks for the least number of distinct multiples of distinct members of an m-set in the first N integers that every interval of length 2N holds; it is min(m, ceiling of 2 root m), by van Doorn, Li and Tang (2026); not root m.

E0675/: Asks which sets of integers are locally periodic, in that membership up to n is unchanged by some shift, for sums of two squares and other examples.

E0677/: Asks whether two disjoint blocks of k consecutive integers can have the same least common multiple; Erdős's 1979 conjecture that they cannot is open, with a few solutions known only when the block lengths differ.

E0678/: Asks whether infinitely often a block of k consecutive integers has a larger least common multiple than a later block of k+1; proved in a strong form by Cambie (2024, arXiv), the ratio exceeding any constant.

E0687/: Asks for the order of the longest initial interval that one residue class per prime up to x can cover, the covering form of Jacobsthal's function; open between x log x times iterated logarithms and Iwaniec's x squared.

E0688/: Asks for the largest exponent such that one residue class per prime between n to that exponent and n covers every integer from 1 to n, and whether it tends to zero; open, between Erdős's lower bound and a counting bound of 1/e.

E0689/: Asks whether, for large n, one residue class per prime up to n can be chosen so that every integer from 1 to n lies in at least two of them; open on the site, with three pending AI-assisted full claims of April and July 2026.

E0691/: Asks for a necessary and sufficient condition on a set of positive integers for its set of multiples to have density one; open, with one family of block sequences settled by Tenenbaum.

E0695/: Asks how fast a chain of primes, each congruent to 1 modulo the previous one, must grow, and whether a nearly optimally slow such chain exists.

E0708/: Bounds the fewest integers from a run of consecutive integers whose product is divisible by the product of n given integers; the corrected at-most statement is open, with a partial bound of 12n pending.

E0709/: The least multiplier f(n) such that any f(n) max(A) consecutive integers hold distinct multiples of the n members of A, for every n-set A; known to lie between log n over log log n and root n, with a formula asked for.

E0710/: Asks for an asymptotic formula for the shortest interval just above n with distinct integers, the kth divisible by k, for k up to n; known between n root log n over log log n and 1.74 n root log n, with a 2026 claim pending.

E0711/: Bounds the shortest interval anywhere holding distinct integers, the kth divisible by k, for k up to n, against the one just above n; the comparison is proved (van Doorn 2026), the n to the 1+o(1) bound open, n^(3/2) proved.

E0726/: Asks whether the sum of one over p, over primes p at most n whose remainder of n lies in the upper half of the interval up to p, is about half of log log n.

E0748/: Asks whether the number of sum-free subsets of 1 up to n is two to the power of half of n times one plus a vanishing term.

E0768/: Asks whether the proportion of n up to N such that every prime factor p of n has a divisor of n above one congruent to one mod p decays like exp(-(c+o(1)) sqrt(log N) log log N).

E0770/: Densities and growth of the smallest endpoint at which the collective gcd of the integer power differences becomes one.

E0771/: The largest size such that, for every m, some subset of one to n of that size has no sub-collection of its elements summing to m.

E0774/: Asks whether every infinite set of natural numbers whose finite subsets each contain a dissociated subset (one with distinct subset sums) of proportional size is a finite union of dissociated sets.

E0783/: Asks which set of pairwise coprime integers between two and N, with reciprocal sum at most a fixed constant, leaves the fewest integers up to N divisible by none of its elements; read, with the site, up to o(N), where the largest primes up to N are optimal.

E0784/: Asks whether a bounded reciprocal sum for a set of divisors forces at least x over a power of log x integers up to x divisible by none of them.

E0786/: Asks whether, for every small epsilon, some set of natural numbers of density above one minus epsilon has equal products only between equally many factors.

E0793/: Asks whether the largest subset of one to n in which no element divides the product of two others has an asymptotic led by a prime-counting term; proved with the constant 27/2 in Chojecki's 2026 manuscript, accepted by the site.

E0795/: Asks whether the largest subset of one to n with all subset products distinct is bounded by the primes up to n plus those up to the square root of n; proved by Raghavan with a power-saving error term.

E0796/: The largest subset of one to n in which every number has fewer than k representations as a product of two distinct members; asks whether the second-order term for k equal to three has an asymptotic constant.

E0820/: Infinitely-often coprimality of two and three power differences, and the growth of the least bases admitting a coprime pair.

E0839/: Asks whether an increasing integer sequence in which no term is a sum of consecutive earlier terms must have terms growing faster than linearly at times.

E0848/: Asks whether the largest set of integers up to N with no two members whose product plus one is squarefree is those congruent to seven mod twenty-five.

E0851/: Asks whether some bounded r makes the integers of the form a power of two plus a number with at most r prime divisors have density at least one minus epsilon.

E0854/: Asks about the arithmetic of the primorials, the products of the first k primes.

E0856/: Estimates, for k at least three, the largest reciprocal sum of a set of integers up to N with no k members sharing the same pairwise least common multiple.

E0873/: Asks whether, for every positive epsilon, some k makes the number of windows of k consecutive members with least common multiple below X less than X to epsilon.

E0879/: Estimates the largest sum of a set of pairwise coprime integers up to n, and asks whether the best such set must contain a number with at least k prime factors.

E0888/: The size of the largest subset of one to n in which any four elements with square product must pair off so the outer product equals the inner product.

E0896/: Estimates, for subsets A and B of {1,...,N}, the largest possible number of integers with exactly one representation ab with a in A and b in B; the order of magnitude is N^2/((log N)^delta (log log N)^(3/2)).

E0929/: Asks for the least prime cutoff x such that a positive density of blocks of k consecutive integers have every member divisible by a prime up to x; the inverse of Problem 687's covering function, open above the square root of k.

E0932/: Asks whether infinitely many prime gaps contain at least two integers all of whose prime factors are smaller than the length of that gap.

E0954/: Concerns the growth of the sequence starting 0 and 1 in which each new term is the least n for which the number of pairwise sums at most n is less than n.

E0961/: Estimates the least n such that every set of n consecutive integers above k contains one divisible by a prime greater than k; open between a Rankin-type prime-gap bound and k over log k times iterated logarithms.

E0962/: Estimates the greatest k for which some run of k consecutive integers starting at most at n has every term divisible by a prime larger than k; log k(n) is between c sqrt(log n log log n) and (log n)/2; both questions open.

E0968/: Asks whether the set of n for which the nth prime divided by n is less than the next such ratio has positive density.

E0970/: Asks for the order of magnitude of Jacobsthal's function over integers with at most k distinct prime factors, and whether it is O(k^2); the quadratic bound is proved by an accepted partial claim, and the order of magnitude stays open.

E0971/: Asks whether the least prime congruent to a modulo d exceeds a fixed factor above Euler's totient of d times log d for many residues a; two 2026 proof claims answering yes, one with a Lean development, pending and unreviewed.

E0980/: Asks whether the sum over primes below x of the least kth power nonresidue is asymptotic to a constant times x over log x; proved for every k under Elliott's convention (Erdős 1961 for k=2, Elliott 1967); the sum over every prime with a kth power nonresidue is an open variant for composite k.

E0983/: The least r such that every k-element subset of the first n integers contains more than r members divisible only by primes from some set of r primes.

E0985/: Asks whether every prime p > 2 has a primitive root modulo p that is itself a prime smaller than p; the site's wording also includes p = 2, where it fails.

E1057/: Asks whether the number of Carmichael numbers up to x is x to the power one minus a quantity tending to zero.

E1062/: The largest subset of one to n in which no element divides two other distinct elements, and whether its density tends to an irrational limit; an exact formula and an irrational limit near 0.67297, by a Lean proof Conjectures.io certified.

E1063/: Estimates the least n at least two k for which n minus i divides n choose k for all but one i below k.

E1073/: Asks whether the number of composite numbers below x dividing n factorial plus one for some n is at most x to a power tending to zero.

E1074/: Asks whether the m for which m factorial plus one has a prime factor not congruent to one modulo m, and the primes arising so (Pillai primes, counted among all primes), have densities, and what they are.

E1101/: Asks whether a good sequence of pairwise coprime integers with convergent reciprocal sum can grow only polynomially, or at most subexponentially.

E1102/: Asks how fast a sequence must increase if, for every n, only finitely many members a make n+a squarefree (property P), or if for infinitely many n every member a<n makes n+a squarefree (property Q).

E1103/: Determines how fast an infinite set of integers must grow if every sum of two of its members is squarefree.

E1109/: Estimates the largest subset of the numbers up to N all of whose pairwise sums are squarefree, and whether its size stays below every fixed power of N.

E1134/: Asks whether the smallest set containing one and closed under tripling plus one, doubling plus one, and sextupling plus one has positive lower density.

E1136/: Asks whether some set of naturals of lower density above one third has no two members, possibly equal, summing to a power of two.

E1146/: Asks a question about essential components, sets whose sum with any other set of Schnirelmann density strictly between zero and one raises that density.

E1149/: Determines the density of integers n whose greatest common divisor with the integer part of n to the power alpha is one, for non-integer positive alpha.

E1181/: Asks whether the least prime not dividing the product of the next roughly log n integers after n is below (1-c)(log n)^2 for some c>0 and all large n; only the trivial bound (1+o(1))(log n)^2 is known.

E1204/: Estimates the smallest largest element, and smallest average, of k integers missing a residue class modulo every prime; the largest element lies between half and once k log k, the average between a quarter and half of it; open.

E1205/: The largest number of congruences guaranteed, over choices of one residue class modulo each n up to x, to be satisfied by every integer up to x; the site's own argument gives log x plus lower-order terms, a pending claim.

E1209/: Asks whether one shift n making every n plus a term of a fast-growing sequence prime forces infinitely many, with squarefree and doubly exponential variants; three questions answered no, three stay open.

E1210/: Asks whether a set of pairwise coprime integers below n has the sum of one over n minus a at most the sum of reciprocals of primes below n plus a constant.

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Problems on sets and sequences of integers defined by divisibility, coprimality, gcd or lcm, or residue conditions, or by avoiding a pattern, asking for their density, counting function, gaps or growth, including covering-congruence and sieve questions.

Site tags routed here: additive combinatorics, number theory, primes, sidon sets, squares.