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$a(n)$ is the smallest $r$ where (concatenation of $n$, $r$ times with itself) $\cdot 10 + 1$ is a prime, or $0$ if no such number exists. The number resulting from concatenating $n$, $r$ times, is $n \cdot \sum_{i=0}^{r-1} (10^d)^i$, where $d$ is the number of digits of $n$. Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero. - _Farideh Firoozbakht_, Jan 07 20
$a(n)$ is the smallest prime which has the form of the concatenation $n, n-1, n-2, \dots, n-k$ for some $k < n$, or $0$ if no such prime exists. Conjecture: There are infinitely many composite numbers $n$ such that $a(n)$ is nonzero. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (check
The sequence is defined by $$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k.$$ If $p$ is a prime with $(p/7) = 1$ and $p = x^2 + 7y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 4x^2 - 2p \pmod{p^2}$. - _Zhi-Wei Sun_, Jul 17 2010 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a74
Conjecture: the sequence A228828 is infinite. 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 available. Sources and p
For even $n$, $a(n+2)$ is the greatest integer such that $a(n+2)/a(n+1) < a(n+1)/a(n)$; for odd $n$, the least integer such that $a(n+2)/a(n+1) > a(n+1)/a(n)$; $a(0) = 4, a(1) = 16$. Conjecture: $a(n) = 4 a(n-1) - a(n-3) + a(n-4)$. - Colin Barker, Feb 16 2012 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b23
The sequence $a(n)$ is the smallest positive integer $x$ such that $\sigma_1(x) \bmod x = n$, or $0$ if no such $x$ exists. At present, the 0 entry for $n = 5$ is only a conjecture. That is, it is conjectured that there is no positive integer $x$ such that $\sigma_1(x) \bmod x = 5$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revisi
The sequence is defined by $$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4.$$ Conjecture: Given $F(n,L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$, then $\lim_{n\to\infty} F(n+1,L)/F(n,L) = (\mathrm{Fibonacci}(L)\sqrt{5} + \mathrm{Lucas}(L))/2$ for $L \ge 0$ where $\mathrm{Fibonacci}(n) = \mathrm{A000045}(n)$ and $\mathrm{Lucas}(n) = \mathrm{A000032}(n)$. M
The sequence $a(n)$ is the trajectory of $103$ under the Reverse and Add! operation carried out in base $3$, written in base $10$. $a(0) = 103$, and $a(n+1) = a(n) + \text{rev}_3(a(n))$. $103$ is conjectured to be the smallest number such that the Reverse and Add! algorithm in base $3$ does not lead to a palindrome. Its trajectory is conjectured to never reach a palindrome. Mat
The sequence $a(n)$ is the least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, if such $k$ exists; otherwise $a(n) = 0$. $(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime. Hence $a(4) = 0$. Conjecture: $a(n) = 0$ if and only if $n = 4$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3
Squares of double factorials: $a(n) = ((2n-1)!!)^2 = (1 \cdot 3 \cdot 5 \cdots (2n-1))^2$. Conjecture 1: For any primitive $2n$-th root $\zeta$ of unity, the permanent of the $2n \times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ coincides with $a(n) = ((2n-1)!!)^2$, where $m(j,k)$ is $(1+\zeta^{j-k})/(1-\zeta^{j-k})$ if $j \neq k$, and $1$ otherwise. - Zhi-Wei Sun, Dec 21 2021 Mathemati
Difference of sums of digits of $n$ in ternary and in binary: $$a(n) = \sum \mathrm{digits}_3(n) - \sum \mathrm{digits}_2(n).$$ "This sequence is positive on average, since 1/log(3) > 1/log(4). Do all integers appear infinitely often?" - Charles R Greathouse IV, Feb 07 2013 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8d
Differences in adjacent elements of the sequence quantifying the steps needed for $n$ to converge to 1 in the Collatz Conjecture. $$a(n) = \mathrm{A006577}(n+1) - \mathrm{A006577}(n)$$ for $n > 0$. Conjecture 1: More than half of the terms are 0. - _Ya-Ping Lu_, May 04 2024 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8d
The sequence $a(n)$ counts the number of prime numbers in the interval $[n^n, n^n + n^2]$: $$a(n) = |\{p \text{ prime} \mid n^n \le p \le n^n + n^2\}|$$ Question: for any $n > 0$, is there at least one prime $p$ such that $n^n \le p \le n^n + n^2$? In this case, that would be stronger than the Schinzel conjecture: "for $m > 1$ there's at least one prime $p$ such that $m \le p \
The sequence $a(n)$ counts the number of primes $p$ in the interval $[2^n, 2^n + p_n]$, where $p_n$ is the $n$-th prime ($p_1 = 2, p_2 = 3, \dots$): $$a(n) = |\{p \text{ prime} \mid 2^n \le p \le 2^n + p_n\}|$$ for $n \ge 1$, and $a(0) = 0$. For any $n > 0$, is there always at least one prime $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$? (checked up to $n = 250$). In t
$a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime. Conjecture 2: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$ for $n, m \ge 1$. - _Ulrich Krug_, 2009 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).
The sequence $a(n)$ is the determinant of the $n \times n$ matrix $M$ defined by $M(i,j) = 1$ if $i^2 + j^2$ is prime, and $0$ otherwise, where $1 \le i, j \le n$. Conjecture: $a(n) = 0$ for no $n > 28$. - _Zhi-Wei Sun_, Aug 26 2013 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checke
$$a(n) = |\{(x, y, z) \in \mathbb{N}^3 : x^2 + p_5(y) + p_6(z) = n\}|$$ In April 2009, _Zhi-Wei Sun_ conjectured that $a(n) > 0$ for every $n = 0, 1, 2, 3, \dots$. 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
Number of ways to write the $n$-th positive odd integer in the form $p + 2^x + 11 \cdot 2^y$ with $p$ a prime congruent to $1 \bmod 6$ and $x, y$ positive integers. $$a(n) = \left|\left\{(p, x, y) : p + 2^x + 11 \cdot 2^y = 2n - 1 \text{ with } p \text{ prime}, p \equiv 1 \pmod 6, x, y \in \mathbb{Z}^+\right\}\right|.$$ On Feb. 24, 2009, Zhi-Wei Sun conjectured that $a(n) = 0$
A56777 lists composite numbers $n$ satisfying both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$. The conjectures state identities connecting A56777 and prime quadruples (A7530), as well as congruences satisfied by the members of A56777. All members of the sequence A56777 come from prime quadruples. Mathematical status Open: marked `research open` in goo
$a(n) = \min \{k \ge 1 \mid \text{Prime}(2 \cdot n^k - 1)\}$ for $n \ge 2$. Is $a(n)$ defined for all $n \ge 2$? That is, does there exist $k > 0$ such that $2 \cdot n^k - 1$ is prime? Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop defini