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Numerator of $1/\det(M)$ where $M$ is the $n \times n$ matrix with $M[i,j] = 1/\operatorname{lcm}(i,j)$. "Conjecture: $1/\det(M)$ is an integer only for n: 1 to 34, 36 and 38. All denominators are powers of two (A000079). - _Robert G. Wilson v_, Aug 02 2015" Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b238
Differences between consecutive primes: $a(n) = p_{n+1} - p_n$. Any subsequence a(n .. n+m) with n > 2 (as to exclude the untypical primes 2 and 3) should occur infinitely many times at other starting points k. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal
$a(n) = \text{numerator}(H_n) - \text{denominator}(H_n)$, where $H_n = 1 + 1/2 + \dots + 1/n$. "Conjecture: for $n > 2$, $n$ divides $a(n-2)$ if and only if $n$ is a prime. Checked up to 20000. - _Amiram Eldar_ and _Thomas Ordowski_, Jul 27 2019" Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4
$a(n)$ is the number of primes with odd binary weight (odious primes) among the first $n$ primes minus the number with even binary weight (evil primes). Shevelev conjectures that $a(n) \ge 0$ for $n > 3$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availa
$a(0) = 1, a(1) = 3$; for $n \ge 2$, $a(n) = f(a(n-1)) + f(a(n-2))$ where $f(x) = x/2$ if $x$ is even and $f(x) = x$ if $x$ is odd. Conjecture (1): The natural density of even terms in the sequence is $1/2$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal ava
$$a(n) = \sum_{k=0}^n \binom{2k}{k}^2$$ Conjecture: For any positive integer n, the polynomials Sum_{k=0}^n binomial(2k,k)^2*x^k and Sum_{k=0}^n binomial(2k,k)^2*x^k/(k+1) are irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 23 2013 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77b
The $n$-th term is the denominator of $\frac{B_{2n}}{2n(2n-1)}$ where $B_{2n}$ is the $2n$-th Bernoulli number. Conjecture I: if $n > 2$, then $\frac{a(\text{A005382}(n))}{12}$ is prime, where A005382 is the sequence of primes $p$ such that $2p-1$ is also prime. - Lorenzo Sauras Altuzarra, Oct 13 2020 Mathematical status Open: marked `research open` in google-deepmind/formal-co
"Conjecture: $a(n) <= \frac{n^2-1}{2}$. - _Aspen A.M. Meissner_, Mar 06 2025" Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier befor
Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than $10^-9$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supp
Apéry numbers: $$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$$ For each $n = 1, 2, 3, \dots$ the polynomial $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$ is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba
Conjecture: For each k = 2,3,..., all the rational numbers $\frac{\sigma_k(n)}{n^k} = \sum_{d|n} \frac{1}{d^k}$ (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A
Since $\sqrt{k(k+1)(k+2)(k+3)+1} = k^2 + 3k + 1$, $a(n) = \#\{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} \}.$ Conjecture: $a(n)/A006880(n) \rightarrow 1.77...$ where A006880(n) is the number of primes $\le 10^n$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4
It is conjectured that every odd number occurs in this sequence. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is avail
$a(n)$ is the numerator of $\sum_{k = 1}^n \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2$ for $n \ge 1$ with $a(0) = 0$. We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$, with a finite number of exceptions that depend on $m$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (
$$a(n) = \min \{p \in \mathbb{P} \mid p \equiv 1 \pmod n\}$$ "Conjecture: $a(n) < n^2$ for $n > 1$. - _Thomas Ordowski_, Dec 19 2016" Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for the pinned Lean 4.33.1 environ
Number of primes strictly less than $n^3$. Conjecture (i): for any integer $k > 2$, the sequence $\pi(n^k)/n^k$ ($n = 2, 3, \ldots$) is strictly decreasing, where $\pi(x)$ denotes the number of primes not exceeding $x$. - Zhi-Wei Sun, Oct 17 2015 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4
Here $\mathrm{prime}(n)$ is the $n$-th prime number, and $\pi(n)$ is the prime-counting function. Conjecture: As $n \rightarrow \infty$, there are infinitely many n's such that $a(n)$ is greater than $a(n+1)$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal a
$a(n) = \mathrm{prime}(2n+2) - \mathrm{prime}(2n+1) - \mathrm{prime}(2n) + \mathrm{prime}(2n-1)$, where $\mathrm{prime}(k)$ is the $k$-th prime number. Do the absolute values cover A004275? A004275 is $1$ together with the nonnegative even numbers. The conjecture asks whether every member of A004275 occurs as $|a(n)|$ for some term of the sequence. Mathematical status Open: mar
Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n. Sierpinski's conjecture (1958) is precisely that $a(n) >= n$ for all $n$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for the pinned Lean 4.33
Moll's conjecture 5.5 extends to this sequence and takes the form: (ii) for the other primes of type $2$, the p-adic valuation $\nu_p(a(n)) \sim n/(p - 1)$ as $n \rightarrow \infty$. (Type 2 primes consists of primes p == 1 (mod 4)) Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checke