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1af18651 |
Sums of two cubes: is the representation count $1_A*1_A(n)\ll(\log n)^{O(1)}$? (Erdős #829) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
6fba68f8 |
Graham's conjecture: infinitely many $n$ with $\phi(n)\mid n+a$ for every $a$? (Erdős #828) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
64350cb5 |
Infinitely many $n$ with $\tau(n+k)\ll k$ for all $k\geq 1$? (Erdős #826) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
c689c7f0 |
Count coprime pairs with equal sum-of-divisors: is $h(x)>x^{2-o(1)}$? (Erdős #824) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
c98d9a74 |
Erdős–Pomerance: asymptotics of the window $(n,n+f(n))$ holding distinct multiples of $1,\ldots,n$ (Erdős #710) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
127598a0 |
Erdős–Surányi distinct multiples in a window: bound $f(n)$ between $\log n/\log\log n$ and $\sqrt n$ (Erdős #709) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
a29f5ba4 |
Erdős–Surányi product divisibility: is $g(n)\leq(2+o(1))n$? (Erdős #708) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
4c95a5df |
Growth of prime chains $p_{i+1}\equiv 1\pmod{p_i}$: is $\lim_k p_k^{1/k}=\infty$? (Erdős #695) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
553bcdfc |
Characterise the Behrend sequences: which $A$ make the set of multiples $M_A$ have density 1? (Erdős #691) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
44cee90e |
Covering $[1,n]$ by residues of only the large primes: estimate $\epsilon_n$; is $\epsilon_n=o(1)$? (Erdős #688) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
8f16e35a |
Estimate the Jacobsthal-type covering maximum $Y(x)$: is $Y(x)=o(x^2)$? (Erdős #687) |
OPEN |
0 inv |
4.0 |
2.0 |
29d ago |
|
20944fcf |
Estimate $n_k$: least $n>2k$ with $(n-1)(n-2)\cdots(n-k)$ having no prime factor in $(k,2k)$ (Erdős #451) |
ACTIVE |
2 inv |
3.0 |
3.0 |
28d ago |
|
e3ce6737 |
For $c>1/2$ and large $p$, does every interval $(n,n+p^c)$ contain $a,b$ with $ab\equiv1\pmod p$? (Erdős #445) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
c9f313ff |
Is $\Lambda(k,3)$ finite for all odd $k$, and how fast do $\Lambda(k,2),\Lambda(k,3)$ grow? (Erdős #436) |
ACTIVE |
2 inv |
3.0 |
3.5 |
24d ago |
|
63c2f652 |
How dense can the sumset $A+B$ be if all its elements are pairwise coprime? (Erdős #432) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
2bd31447 |
For large $n$, must the greedy $[1,n)$ sequence with all prime factors $>n-a$ include a composite? (Erdős #430) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
15ab61af |
Does the sequence built from $2,3$ by adjoining all $a_ia_j-1$ have positive density? (Erdős #424) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
9f8d9815 |
Growth of the greedy sequence whose terms are the least new sum of $\ge 2$ consecutive earlier terms (Erdős #423) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3ec60c1b |
Does the Hofstadter Q-sequence $f(n)=f(n-f(n-1))+f(n-f(n-2))$ miss infinitely many integers? (Erdős #422) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
6a8e8519 |
Density and growth of $\tau((n+f(n))!)/\tau(n!)$, ratios of divisor-counts of nearby factorials (Erdős #420) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
160d8891 |
Two ways to count Euler-totient values: does $V(x)/V'(x)$ converge, and does it exceed 1? (Erdős #417) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
9e60feb0 |
Distribution of Euler-totient values: does $V(2x)/V(x)\to 2$, with an asymptotic for $V(x)$? (Erdős #416) |
OPEN |
0 inv |
3.5 |
2.5 |
36d ago |
|
16efa709 |
Summatory growth of $t_k(n)$, the least start making $n$ divide a run of $k$ consecutive integers (Erdős #394) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
03500b7e |
Does every n admit a k with the product of k consecutive integers from n dividing the next k? (Erdős #389) |
OPEN |
0 inv |
2.5 |
3.5 |
36d ago |
|
63ce256d |
Are there only finitely many equal products of two disjoint blocks of 4+ consecutive integers? (Erdős #388) |
ACTIVE |
1 inv |
2.5 |
3.0 |
15d ago |
|
5cc91e89 |
Is $\max_{m<n}(m+p(m))>n$ eventually and does the excess diverge? (Erdős #385) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
1dc57ca1 |
Infinitely many primes $p$ with top prime factor of $\prod_{0\le i\le k}(p^2+i)$ equal to $p$? (Erdős #383) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
a9c6ac49 |
Runs of consecutive integers whose product's top prime is squared: can $v-u$ be unbounded? (Erdős #382) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
316f41fc |
How fast does $|D_k\cap[1,n]|$ grow for the factorial-product-square index $F(m)=k$? (Erdős #374) |
OPEN |
0 inv |
3.0 |
4.0 |
36d ago |
|
528b3173 |
Does $\{n : P(n)<P(n+1)\}$ have natural density exactly $1/2$? (Erdős #371) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
df853a01 |
Growth of $F(n)$, the largest prime factor of $n(n+1)$: how small can it be? (Erdős #368) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
96f33311 |
Bound the product of the 2-full parts of $k$ consecutive integers: is it $n^{2+o(1)}$? (Erdős #367) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
81bb3dac |
Bound the number of consecutive powerful pairs up to $x$: is it $(\log x)^{O(1)}$? (Erdős #365) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
fc8fe966 |
Largest subset of $\{1,\ldots,\lfloor cn\rfloor\}$ having no subset summing to $n$ (Erdős #361) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
f5fdaa68 |
Density and growth of MacMahon's prime numbers of measurement (segmented numbers) (Erdős #359) |
OPEN |
0 inv |
2.5 |
3.0 |
36d ago |
|
5e28fa54 |
Growth of $f(n)$: largest increasing set in $[n]$ with all consecutive-block sums distinct (Erdős #357) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
7a5c5cc0 |
Completeness of $\{\lfloor 2^k\alpha\rfloor\}\cup\{\lfloor 2^k\beta\rfloor\}$ for irrational $\alpha/\beta$ (Erdős #354) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
0bf09014 |
Ulam numbers: twin pairs, eventual gap-periodicity, and zero density (Erdős #342) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
8017d237 |
Eventual periodicity of the gaps of Dickson's greedy sum-avoiding sequence (Erdős #341) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3633f94b |
Best smoothness function $f(n)$ writing every $n$ as a sum of two $f(n)$-smooth integers (Erdős #334) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
c29b53eb |
Sufficient conditions for the infinitely-recurring difference set $D(A)$ to have bounded gaps (Erdős #332) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
f2bf4f3a |
How dense can an infinite Sidon set be along N^{1/2}? (Erdős #329) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
cf4e1d54 |
Is n/2^n always a finite sum of distinct terms a/2^a? (Erdős #261) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
33e9b8e2 |
A density and equidistribution condition forcing subset-sum completeness (Erdős #254) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
994de563 |
Must a near-squaring integer sequence with rational reciprocal sum be Sylvester's sequence? (Erdős #243) |
OPEN |
0 inv |
3.0 |
1.0 |
36d ago |
|
7e911005 |
How large can gaps between consecutive squarefree numbers be? (Erdős #208) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
e8c1aa10 |
Do all power-moments of gaps between consecutive squarefree numbers converge? (Erdős #145) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
2f4779cc |
Can a product of k ≥ 3 consecutive integers ever be powerful? (Erdős #137) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
1443d057 |
Estimate the maximum size of a non-dividing subset of {1,...,N} (Erdős #131) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3c3bbdb0 |
Do all orbits of $n\mapsto n+\tau(n)$ eventually merge into one sequence? (Erdős #414) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |