Problems · OpenFrontiers

OP-9833Number theoryStatus unassessed

Denominator of sum of reciprocals of divisors

Formal Conjectures· Specialist source · status unassessed

OP-9832Number theoryStatus unassessed

Recurrence with fourth powers of binomial coefficients

Formal Conjectures· Specialist source · status unassessed

OP-9831Number theoryStatus unassessed

Smallest index $k > n$ such that $(p_k+p_{k+1})/(p_n+p_{n+1})$ is an integer $\ge 2$

Smallest index k n such that (p k+p k+1 )/(p n+p n+1 ) is an integer ge 2

Formal Conjectures· Specialist source · status unassessed

OP-9830Number theoryStatus unassessed

Chua's Euclidean prime sequence

Formal Conjectures· Specialist source · status unassessed

OP-9829Number theoryStatus unassessed

Rowland-style prime-generating recurrence

Formal Conjectures· Specialist source · status unassessed

OP-9828Number theoryStatus unassessed

Number of ways to express $n$ as sum of square, pentagonal, and hexagonal numbers

Number of ways to express n as sum of square, pentagonal, and hexagonal numbers

Formal Conjectures· Specialist source · status unassessed

OP-9827Number theoryStatus unassessed

Smallest $m$ such that $n^3 + m^3 + 1$ is prime

Smallest m such that n^3 + m^3 + 1 is prime

Formal Conjectures· Specialist source · status unassessed

OP-9826Number theoryStatus unassessed

Representations as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$

Representations as p + 2^x + 11 cdot 2^y with p equiv 1 pmod 6

Formal Conjectures· Specialist source · status unassessed

OP-9825Number theoryStatus unassessed

Collatz step differences

Formal Conjectures· Specialist source · status unassessed

OP-9824Number theoryStatus unassessed

Factorial distance to nearest square

Formal Conjectures· Specialist source · status unassessed

OP-9823Number theoryStatus unassessed

Number of Abelian cubes of length $3n$ over an alphabet of size 3

Number of Abelian cubes of length 3n over an alphabet of size 3

Formal Conjectures· Specialist source · status unassessed

OP-9822Number theoryStatus unassessed

Lesser of twin primes

Formal Conjectures· Specialist source · status unassessed

OP-9821Number theoryStatus unassessed

Recurrence involving LCM: $a(n) = x(n+1)/x(n) - 2$

Recurrence involving LCM: a(n) = x(n+1)/x(n) - 2

Formal Conjectures· Specialist source · status unassessed

OP-9820Number theoryStatus unassessed

Odious primes minus evil primes among first $n$ primes

Odious primes minus evil primes among first n primes

Formal Conjectures· Specialist source · status unassessed

OP-9819Number theoryStatus unassessed

Prime gaps

Formal Conjectures· Specialist source · status unassessed

OP-9818Number theoryStatus unassessed

Half-Fibonacci sequence

Formal Conjectures· Specialist source · status unassessed

OP-9817Number theoryStatus unassessed

Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime

Least k ge 1 such that 2 cdot n^k - 1 is prime

Formal Conjectures· Specialist source · status unassessed

OP-9816Number theoryStatus unassessed

Sum of Fermat number and Mersenne number minus 1: $2^{2^n} + 2^n - 1$

Sum of Fermat number and Mersenne number minus 1: 2^ 2^n + 2^n - 1

Formal Conjectures· Specialist source · status unassessed

OP-9815Number theoryStatus unassessed

Least $k$ such that cyclotomic polynomial $\Phi_k(n)$ is prime

Least k such that cyclotomic polynomial Phi k(n) is prime

Formal Conjectures· Specialist source · status unassessed

OP-9814Number theoryStatus unassessed

Number of primes in $n$ -th row of triangle $k^2 - k + p_n$

Number of primes in n -th row of triangle k^2 - k + p n

Formal Conjectures· Specialist source · status unassessed

OP-9813Number theoryStatus unassessed

Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes

Formal Conjectures· Specialist source · status unassessed

OP-9812Number theoryStatus unassessed

$a(n) = \sum_{j=1}^{n} (3^j + (-2)^j)$

a(n) = sum j=1 ^ n (3^j + (-2)^j)

Formal Conjectures· Specialist source · status unassessed

OP-9811Number theoryStatus unassessed

Sum of squares of divisors of $n$

Sum of squares of divisors of n

Formal Conjectures· Specialist source · status unassessed

OP-9810Number theoryStatus unassessed

$a(n)$ is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$

a(n) is the integer whose decimal digits are the first n+1 decimal digits of pi

Formal Conjectures· Specialist source · status unassessed

OP-9809Number theoryStatus unassessed

$a(n)$ = the number of values of $k <= 10^n$ such that $\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime

a(n) = the number of values of k <= 10^n such that sqrt k(k+1)(k+2)(k+3)+1 is prime

Formal Conjectures· Specialist source · status unassessed

Denominator of sum of reciprocals of divisors

Recurrence with fourth powers of binomial coefficients

Smallest index k>nk > nk>n such that (pk+pk+1)/(pn+pn+1)(p_k+p_{k+1})/(p_n+p_{n+1})(pk​+pk+1​)/(pn​+pn+1​) is an integer ≥2\ge 2≥2

Chua's Euclidean prime sequence

Rowland-style prime-generating recurrence

Number of ways to express nnn as sum of square, pentagonal, and hexagonal numbers

Smallest mmm such that n3+m3+1n^3 + m^3 + 1n3+m3+1 is prime

Representations as p+2x+11⋅2yp + 2^x + 11 \cdot 2^yp+2x+11⋅2y with p≡1(mod6)p \equiv 1 \pmod 6p≡1(mod6)

Collatz step differences

Factorial distance to nearest square

Number of Abelian cubes of length 3n3n3n over an alphabet of size 3

Lesser of twin primes

Recurrence involving LCM: a(n)=x(n+1)/x(n)−2a(n) = x(n+1)/x(n) - 2a(n)=x(n+1)/x(n)−2

Odious primes minus evil primes among first nnn primes

Prime gaps

Half-Fibonacci sequence

Least k≥1k \ge 1k≥1 such that 2⋅nk−12 \cdot n^k - 12⋅nk−1 is prime

Sum of Fermat number and Mersenne number minus 1: 22n+2n−12^{2^n} + 2^n - 122n+2n−1

Least kkk such that cyclotomic polynomial Φk(n)\Phi_k(n)Φk​(n) is prime

Number of primes in nnn-th row of triangle k2−k+pnk^2 - k + p_nk2−k+pn​

Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes

a(n)=∑j=1n(3j+(−2)j)a(n) = \sum_{j=1}^{n} (3^j + (-2)^j)a(n)=∑j=1n​(3j+(−2)j)

Sum of squares of divisors of nnn

a(n)a(n)a(n) is the integer whose decimal digits are the first n+1n+1n+1 decimal digits of π\piπ

a(n)a(n)a(n) = the number of values of k<=10nk <= 10^nk<=10n such that k(k+1)(k+2)(k+3)+1\sqrt{k(k+1)(k+2)(k+3)+1}k(k+1)(k+2)(k+3)+1​ is prime

Smallest index $k > n$ such that $(p_k+p_{k+1})/(p_n+p_{n+1})$ is an integer $\ge 2$

Number of ways to express $n$ as sum of square, pentagonal, and hexagonal numbers

Smallest $m$ such that $n^3 + m^3 + 1$ is prime

Representations as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$

Number of Abelian cubes of length $3n$ over an alphabet of size 3

Recurrence involving LCM: $a(n) = x(n+1)/x(n) - 2$

Odious primes minus evil primes among first $n$ primes

Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime

Sum of Fermat number and Mersenne number minus 1: $2^{2^n} + 2^n - 1$

Least $k$ such that cyclotomic polynomial $\Phi_k(n)$ is prime

Number of primes in $n$ -th row of triangle $k^2 - k + p_n$

$a(n) = \sum_{j=1}^{n} (3^j + (-2)^j)$

Sum of squares of divisors of $n$

$a(n)$ is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$

$a(n)$ = the number of values of $k <= 10^n$ such that $\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime