Problems · OpenFrontiers

OP-9908Number theoryStatus unassessed

$a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$

a(1)=1, a(2)=1, a(n)=(2^ a(n-1) + 1)/a(n-2)

Formal Conjectures· Specialist source · status unassessed

OP-9907Number theoryStatus unassessed

Trajectory of 103 under the Reverse and Add! operation in base 3

Formal Conjectures· Specialist source · status unassessed

OP-9906Number theoryStatus unassessed

Smallest $x$ such that $\sigma(x) \bmod x = n$

Smallest x such that sigma(x) bmod x = n

Formal Conjectures· Specialist source · status unassessed

OP-9905Number theoryStatus unassessed

Number of times $n$ occurs as a binary sub-pattern of $n^2$

Number of times n occurs as a binary sub-pattern of n^2

Formal Conjectures· Specialist source · status unassessed

OP-9904Number theoryStatus unassessed

Sum of next $n$ primes

Sum of next n primes

Formal Conjectures· Specialist source · status unassessed

OP-9903Number theoryStatus unassessed

Wolstenholme numbers

Formal Conjectures· Specialist source · status unassessed

OP-9902Number theoryStatus unassessed

Smallest factorial containing exactly $n$ 6's

Smallest factorial containing exactly n 6's

Formal Conjectures· Specialist source · status unassessed

OP-9901Number theoryStatus unassessed

Alternating sum of signs of powers of $3/2$

Alternating sum of signs of powers of 3/2

Formal Conjectures· Specialist source · status unassessed

OP-9900Number theoryStatus unassessed

Determinant of matrix with entries indicating primality of $i^2 + j^2$

Determinant of matrix with entries indicating primality of i^2 + j^2

Formal Conjectures· Specialist source · status unassessed

OP-9899Number theoryStatus unassessed

Value of $n$ -th cyclotomic polynomial at $n$

Value of n -th cyclotomic polynomial at n

Formal Conjectures· Specialist source · status unassessed

OP-9898Number theoryStatus unassessed

Catalan-Mersenne numbers

Formal Conjectures· Specialist source · status unassessed

OP-9897Number theoryStatus unassessed

Number of primes $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$

Number of primes p such that 2^n le p le 2^n + mathrm prime (n)

Formal Conjectures· Specialist source · status unassessed

OP-9896Number theoryStatus unassessed

Number of primes $p$ such that $n^n \le p \le n^n + n^2$

Number of primes p such that n^n le p le n^n + n^2

Formal Conjectures· Specialist source · status unassessed

OP-9895Number theoryStatus unassessed

Number of times $n^2 + s^2$ is prime for positive integers $s < n$

Number of times n^2 + s^2 is prime for positive integers s < n

Formal Conjectures· Specialist source · status unassessed

OP-9894Number theoryStatus unassessed

Sum of $a(k)/k!$ over divisors equals harmonic number

Sum of a(k)/k! over divisors equals harmonic number

Formal Conjectures· Specialist source · status unassessed

OP-9893Number theoryStatus unassessed

Conjectures associated with A067720

Formal Conjectures· Specialist source · status unassessed

OP-9892Number theoryStatus unassessed

Decimal encoding of the prime factorization of $n$

Decimal encoding of the prime factorization of n

Formal Conjectures· Specialist source · status unassessed

OP-9891Geometry and topologyStatus unassessed

Integer part of area of a regular polygon with $n$ sides each of length 1

Integer part of area of a regular polygon with n sides each of length 1

Formal Conjectures· Specialist source · status unassessed

OP-9890Number theoryStatus unassessed

Numerator - denominator in $n$ -th harmonic number

Numerator - denominator in n -th harmonic number

Formal Conjectures· Specialist source · status unassessed

OP-9889Number theoryStatus unassessed

Conjectures associated with A063880

Formal Conjectures· Specialist source · status unassessed

OP-9888CombinatoricsStatus unassessed

Number of different products of subsets of $\{1, 2, \dots, n\}$

Number of different products of subsets of 1, 2, dots, n

Formal Conjectures· Specialist source · status unassessed

OP-9887Number theoryStatus unassessed

Numerator of $1/\det(M)$ for $M[i,j] = 1/\operatorname{lcm}(i,j)$

Numerator of 1/ det(M) for M[i,j] = 1/ operatorname lcm (i,j)

Formal Conjectures· Specialist source · status unassessed

OP-9886Number theoryStatus unassessed

Divisibility of $2^n + 1$ by $n$

Divisibility of 2^n + 1 by n

Formal Conjectures· Specialist source · status unassessed

OP-9885Number theoryStatus unassessed

Least $m$ such that $\phi(m) = n!$

Least m such that phi(m) = n!

Formal Conjectures· Specialist source · status unassessed

OP-9884Number theoryStatus unassessed

Catalan-Larcombe-French sequence

Formal Conjectures· Specialist source · status unassessed

a(1)=1,a(2)=1,a(n)=(2a(n−1)+1)/a(n−2)a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)a(1)=1,a(2)=1,a(n)=(2a(n−1)+1)/a(n−2)

Trajectory of 103 under the Reverse and Add! operation in base 3

Smallest xxx such that σ(x) mod x=n\sigma(x) \bmod x = nσ(x)modx=n

Number of times nnn occurs as a binary sub-pattern of n2n^2n2

Sum of next nnn primes

Wolstenholme numbers

Smallest factorial containing exactly nnn 6's

Alternating sum of signs of powers of 3/23/23/2

Determinant of matrix with entries indicating primality of i2+j2i^2 + j^2i2+j2

Value of nnn-th cyclotomic polynomial at nnn

Catalan-Mersenne numbers

Number of primes ppp such that 2n≤p≤2n+prime(n)2^n \le p \le 2^n + \mathrm{prime}(n)2n≤p≤2n+prime(n)

Number of primes ppp such that nn≤p≤nn+n2n^n \le p \le n^n + n^2nn≤p≤nn+n2

Number of times n2+s2n^2 + s^2n2+s2 is prime for positive integers s<ns < ns<n

Sum of a(k)/k!a(k)/k!a(k)/k! over divisors equals harmonic number

Conjectures associated with A067720

Decimal encoding of the prime factorization of nnn

Integer part of area of a regular polygon with nnn sides each of length 1

Numerator - denominator in nnn-th harmonic number

Conjectures associated with A063880

Number of different products of subsets of {1,2,…,n}\{1, 2, \dots, n\}{1,2,…,n}

Numerator of 1/det⁡(M)1/\det(M)1/det(M) for M[i,j]=1/lcm⁡(i,j)M[i,j] = 1/\operatorname{lcm}(i,j)M[i,j]=1/lcm(i,j)

Divisibility of 2n+12^n + 12n+1 by nnn

Least mmm such that ϕ(m)=n!\phi(m) = n!ϕ(m)=n!

Catalan-Larcombe-French sequence

$a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$

Smallest $x$ such that $\sigma(x) \bmod x = n$

Number of times $n$ occurs as a binary sub-pattern of $n^2$

Sum of next $n$ primes

Smallest factorial containing exactly $n$ 6's

Alternating sum of signs of powers of $3/2$

Determinant of matrix with entries indicating primality of $i^2 + j^2$

Value of $n$ -th cyclotomic polynomial at $n$

Number of primes $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$

Number of primes $p$ such that $n^n \le p \le n^n + n^2$

Number of times $n^2 + s^2$ is prime for positive integers $s < n$

Sum of $a(k)/k!$ over divisors equals harmonic number

Decimal encoding of the prime factorization of $n$

Integer part of area of a regular polygon with $n$ sides each of length 1

Numerator - denominator in $n$ -th harmonic number

Number of different products of subsets of $\{1, 2, \dots, n\}$

Numerator of $1/\det(M)$ for $M[i,j] = 1/\operatorname{lcm}(i,j)$

Divisibility of $2^n + 1$ by $n$

Least $m$ such that $\phi(m) = n!$