Arithmetic Functions

../

E0048/: Asks whether there are infinitely many pairs of integers for which Euler's totient of one equals the sum of the divisors of the other.

E0050/: Asks whether the density of integers whose totient is below a given fraction of the integer, as a function of that fraction, ever has a positive derivative.

E0051/: Asks whether some infinite set of totient values has the smallest integer attaining each value growing faster than any fixed multiple of the value.

E0122/: Characterizes which number theoretic functions f make the shifted values n plus f of n cluster unboundedly in short intervals infinitely often.

E0126/: Asks whether the least number of distinct prime factors of the product of the sums of two distinct elements of a set of n natural numbers grows faster than the logarithm of n.

E0205/: Asks whether every large integer is a power of two plus a number whose count of prime factors with multiplicity is below the iterated logarithm.

E0239/: Asks whether every multiplicative function taking only the values plus and minus one has a mean value.

E0248/: Asks whether there are infinitely many n for which the number of distinct prime factors of n plus k stays of order k for every positive k.

E0334/: The best function f for which every n is a sum of two integers having no prime factor larger than f of n; Erdős asked whether n^epsilon suffices, still open, and whether even n^(1/3) does, which Balog's bound answers yes.

E0367/: Asks whether the product of the powerful parts of k consecutive integers near n is at most about n squared, for every fixed k.

E0368/: Asks how large the largest prime factor of the product of n and n plus one is.

E0369/: Asks whether every large n has k consecutive n^epsilon-smooth integers up to n; trivially true as worded, it is proved in both nontrivial readings, each member smooth to its own power or the run inside [n/2, n].

E0370/: Asks whether infinitely many integers n have both n and n plus one with largest prime factor below their own square root.

E0371/: Asks whether the integers n whose largest prime factor is smaller than that of n plus one have density one half.

E0372/: Asks whether infinitely many integers n have the largest prime factors of n, n plus one, and n plus two in strictly decreasing order.

E0380/: Asks whether the integers up to x lying in an interval whose product has a repeated largest prime factor are asymptotically as many as the n up to x divisible by the square of their own largest prime factor.

E0382/: Asks how long an interval of integers can be when the largest prime dividing its product appears at least twice, and whether that length can be arbitrarily large.

E0383/: Asks whether, for every k, infinitely many primes p are the largest prime factor of the product of p squared through p squared plus k.

E0385/: Asks whether the largest value of a composite number below n plus its least prime factor exceeds n for all large n, and whether the excess grows without bound.

E0408/: Asks whether the number of totient iterations needed to reach one, divided by the logarithm of n, has a distribution or is almost always constant.

E0409/: Asks how many iterations of the map sending n to Euler's totient of n plus one are needed to reach a prime, and how often a given prime is reached.

E0410/: Asks whether the k-th iterate of the sum-of-divisors function, taken to the power one over k, tends to infinity for every integer n at least two.

E0411/: Determines for which n and r the iterates of the map sending n to n plus Euler's totient of n satisfy that shifting by r steps eventually doubles the value.

E0412/: Asks whether, for all m and n at least two, the iterated sum-of-divisors sequences starting at m and at n eventually share a value.

E0413/: Asks whether infinitely many n have the property that every smaller m satisfies m plus its number of distinct prime factors being at most n.

E0414/: Asks whether the iterates of the map sending n to n plus its number of divisors, started from any two integers, always eventually meet at a common value.

E0415/: Estimates the largest k such that every ordering pattern of k consecutive values of Euler's totient function occurs below n, and which pattern fails first.

E0416/: Asks whether the count of totient values up to x doubles when x doubles, and whether that count has an asymptotic formula; the doubling limit is proved twice in Lean, and the release's asymptotic equivalent is a pending claim.

E0417/: Asks whether the ratio of the count of totient values below x to the number of distinct totients of integers below x has a limit exceeding one; the limit's existence is open.

E0418/: Asks whether infinitely many positive integers are not of the form n minus Euler's totient of n.

E0420/: Asks whether the ratio of the divisor counts of the factorials of n plus a power of the logarithm of n and of n tends to infinity for large exponents.

E0452/: The longest interval inside x to twice x on which every integer has more than log log n distinct prime factors.

E0456/: Compares the least prime congruent to one modulo n with the least integer whose Euler totient is divisible by n; a September 2026 forum claim answers the three questions no, no and yes, pending review.

E0491/: Asks whether an additive function whose consecutive differences stay bounded must equal a constant multiple of the logarithm plus a bounded error.

E0520/: Asks whether Rademacher random multiplicative sums over sqrt(N log log N) almost surely have a positive constant limit superior; a Lean proof built and audited here makes the ratio tend to 0, so the answer is no.

E0647/: Asks whether some n greater than 24 has m plus the number of divisors of m at most n plus two for every m less than n.

E0648/: Estimates the length of the longest chain of integers below n whose greatest prime factors are strictly decreasing.

E0649/: Asks whether for any two primes p and q there is an integer n whose greatest prime factor is p while that of n plus one is q.

E0663/: Asks whether the least prime not dividing the product of k consecutive integers above n is at most about the logarithm of n, for fixed k and large n.

E0679/: Asks whether infinitely many n have every n minus k with fewer distinct prime factors than about the logarithm of k over its own logarithm.

E0690/: The density of the integers whose kth smallest prime factor is a given prime p, and how that density behaves as k and p vary.

E0694/: Estimates how large the ratio of the largest to the smallest integer with a given value of Euler's totient function can be for values up to x.

E0821/: Asks whether, for every positive epsilon, infinitely many n have more than n to the power one minus epsilon integers whose Euler totient equals n; a proof is claimed by the OpenAI release of September 2026.

E0822/: Asks whether the integers of the form n plus the Euler totient of n have positive lower density.

E0823/: Asks whether, for every real number at least one, there are pairs of integers with equal sums of divisors whose ratio tends to that number.

E0824/: Estimates the number of coprime pairs of integers below x that have the same sum of divisors; a lower bound with exponent 13/8 is claimed in an AI-assisted write-up of August 2026.

E0825/: Asks whether there is a constant C such that every integer whose sum of divisors exceeds C times it is a sum of distinct proper divisors of itself.

E0826/: Asks whether there are infinitely many n for which the number of divisors of n plus k is at most a constant times k for every k at least one.

E0828/: Asks whether, for every integer a, there are infinitely many n whose Euler totient divides n plus a.

E0830/: Asks whether there are infinitely many amicable pairs, and whether the count of them up to x is at least x to the power one minus a small amount.

E0878/: Compares f(n), the sum over the primes p dividing n of the largest power of p not exceeding n, with F(n), the largest sum of distinct pairwise coprime integers from 2 to n built from those primes.

E0889/: Concerns the number of prime factors of n plus k that exceed k, that is, those dividing none of the earlier terms of the interval.

E0891/: Asks whether, for each k at least 2 and every large enough n, the interval of length p_1...p_k starting at n contains an integer with more than k distinct prime factors.

E0897/: Asks whether an additive function whose values on prime powers are unboundedly large compared with the logarithm must have consecutive differences unboundedly large compared with log n, or even unbounded ratios.

E0928/: Asks whether the density exists of integers n whose largest prime factor is below n to the alpha while that of n plus 1 is below n plus 1 to the beta; the OpenAI release of September 2026 shows it is a Dickman product, accepted on its built Lean proof.

E0955/: Asks whether every density-zero set has a density-zero preimage under the sum of proper divisors; known results cover the primes, sums of two squares, prime-factor-count tails, palindromes, sparse and missing-digit targets.

E0976/: Estimates the greatest prime factor of the product of an irreducible polynomial's values up to n, in particular whether it exceeds n to a power above one.

E0977/: Asks whether the greatest prime factor of two to the n minus one, divided by n, tends to infinity.

E1003/: Asks whether Euler's totient function takes the same value at n and at n plus one for infinitely many n.

E1004/: Asks whether, for every fixed positive c and all large x, some n up to x has all totient values on an interval of length a power of the logarithm of x distinct.

E1052/: Concerns unitary perfect numbers, integers equal to the sum of their proper divisors d for which d and n divided by d are coprime.

E1053/: Asks whether every k with the sum of divisors of n equal to k times n must be of order smaller than log log n.

E1060/: Bounds the number f(n) of integers k with k times the sum of divisors of k equal to n, asking whether f(n) is at most n to a power o(1/log log n), perhaps even a power of log n.

E1061/: Counts pairs with the sum of divisors of a plus that of b equal to that of a plus b and a plus b at most x, and asks whether the count is proportional to x; Li's June 2026 preprint claims it exceeds x times any log power, pending.

E1064/: Asks whether the totient of n exceeds the totient of n minus its totient for almost all n and the reverse holds infinitely often; the first part proved by Luca and Pomerance (2002), the second by Grytczuk, Luca and Wójtowicz (2001).

E1106/: Asks whether the number of distinct prime factors of the product of the partition numbers up to n tends to infinity, and eventually exceeds n.

E1122/: Asks whether an additive function that decreases from n to n plus one only for a density-zero set of n must be a constant multiple of the logarithm.

E1144/: Asks whether a random completely multiplicative sign function almost surely has partial sums up to N exceeding any multiple of the square root of N.

E1203/: Asks whether the maximum over k of the number of distinct prime factors of n plus k times log log k over log k tends to infinity as n grows.

---

Problems on the values, distribution and iterates of named arithmetic functions — Euler's totient, the sum and number of divisors, counts of prime factors, the largest prime factor — along with general additive and multiplicative functions and perfect, amicable and aliquot-type questions.

Site tags routed here: divisors, iterated functions, number theory, powerful, probability, unit fractions.