OEIS

The 346 problem files in FormalConjectures/OEIS that state a research or textbook statement, at commit 2055e5ad, one article each.

Their docstrings and Lean statements are quoted from Formal Conjectures (Google DeepMind) at that commit, under its Apache-2.0 license.

Density of ones in the… to Primes congruent to ${3, 5…

• Density of ones in the Kolakoski sequence
• The prime numbers
• No powers as partition numbers
• Fractional parts of sums of reciprocals of Catalan numbers \(C(n)\)
• Number of squares \(\bmod n\)
• Euclid-Mullin sequence
• \(a(n) = 2^(2^n)\)
• Sum of squares of divisors of \(n\)
• Prime gaps
• Lesser of twin primes
• Ore's harmonic numbers
• Squares of double factorials
• Multiplicative order of 2 mod \(2n+1\)
• Cuban Primes
• Central trinomial coefficients
• Central factorial numbers: \(((2n)!!)^2\)
• Coefficient of \((xyz)^n\) in \(((x+y)(y+z)(z+x))^n\) equaling \(\binom{2n}{n}^3\)
• A binomial coefficient sum
• A binomial coefficient summation
• Primes congruent to \(\{3, 5, 6\} \pmod 7\)

Least positive multiple of \(n\)… to Number of primes \(< n^2\)

• Least positive multiple of \(n\) in base 10 with digits 0 and 1
• Recamán's sequence
• Practical numbers
• Apéry numbers
• Subword complexity of the morphism a → aab, b → b
• Catalan-Mersenne numbers
• Wolstenholme numbers
• Sum of next \(n\) primes
• Least prime \(\ge n\)
• Prime numbers at the beginning of the 20th century
• \(a(n)\) is the integer whose decimal digits are the first \(n+1\) decimal digits of \(\pi\)
• Denominator of sum of reciprocals of divisors
• A nonlinear recurrence sequence
• Determinant of Hankel matrix of the first \(2n-1\) prime numbers
• Combinatorial representation of the recurrence \(a(n+2) = 2a(n+1) + 2a(n)\)
• Smallest number \(k\) such that \(kn + 1\) is prime
• Smallest prime \(\equiv 1 \pmod n\)
• Home primes (OEIS A037274)
• Number of primes \(< n^3\)
• Number of primes \(< n^2\)

Conjectures associated with… to Decimal encoding of the prime…

• Conjectures associated with A038552
• Smallest composite \(c\) such that \(\textrm{primorial}(n) + c\) is prime
• Denominators of coefficients in Stirling's expansion for \(\log(\Gamma(z))\)
• \(a(n) = \sum_{k=1}^n (k^2 \bmod n)\)
• Nearest integer to \(n/\sqrt{2}\)
• Asymptotics of subsets of \(\{1, 2, \dots, n\}\) with integer average
• Maximum exponent in the prime factorization of \(n\)
• Expansion of g.f. \((1-\sqrt{1-4x-4x^2})/(2(1+x))\)
• \(a(n) = (\text{smallest prime} > n^2) - n^2\)
• Concatenation of the next \(n\) numbers
• Catalan-Larcombe-French sequence
• Least \(m\) such that \(\phi(m) = n!\)
• Divisibility of \(2^n + 1\) by \(n\)
• Numerator of \(1/\det(M)\) for \(M\[i,j\] = 1/\operatorname{lcm}(i,j)\)
• Number of different products of subsets of \(\{1, 2, \dots, n\}\)
• Reversible multiples \(a(3^n) = 10^{3^{n-2}} - 1\) for powers of \(3\)
• Conjectures associated with A063880
• Numerator - denominator in \(n\)-th harmonic number
• Integer part of area of a regular polygon with \(n\) sides each of length 1
• Decimal encoding of the prime factorization of \(n\)

Conjectures associated with… to Number of prime powers strictly…

• Conjectures associated with A067720
• Euclid–Fortunate prime coverage
• Sum of \(a(k)/k!\) over divisors equals harmonic number
• Number of times \(n^2 + s^2\) is prime for positive integers \(s < n\)
• Determinant of prime sum indicator matrix
• Number of primes \(p\) such that \(n^n \le p \le n^n + n^2\)
• Number of primes \(p\) such that \(2^n \le p \le 2^n + \mathrm{prime}(n)\)
• Value of \(n\)-th cyclotomic polynomial at \(n\)
• Recurrence \(a(n+2) = |a(n+1)a(n) - a(n)a(n+1)|\) via concatenation
• Determinant of matrix with entries indicating primality of \(i^2 + j^2\)
• Alternating sum of signs of powers of \(3/2\)
• Smallest factorial containing exactly \(n\) 6's
• Number of times \(n\) occurs as a binary sub-pattern of \(n^2\)
• Smallest \(x\) such that \(\sigma(x) \bmod x = n\)
• Trajectory of 103 under the Reverse and Add! operation in base 3
• \(a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)\)
• Smallest \(m > 0\) such that \(n \cdot 2^m + 1\) is prime
• Least \(k > 0\) such that \((k+1)(k+2)\cdots(k+n) + 1\) is prime
• \(a(n) = \sum_{k=0}^n \binom{2k}{k}^3\)
• Number of prime powers strictly between \(n\)-th prime and \((n+1)\)-th prime

Conjecture relating two… to Prime-th recurrence with…

• Conjecture relating two characterizations of a set of integers.
• Denominator of \(\sum_{k=1}^n k^{\mu(k)}\)
• Primes of the form 2^n + 2^i + 1
• Smallest palindrome with exactly \(n\) divisors
• Smallest prime \(p\) such that \(p + n\) is an \(n\)-th power
• Smallest \(r\) such that (concatenation of \(n\), \(r\) times) \(\cdot 10 + 1\) is prime
• Binary representation of primes that divide a number, in decimal
• Expansion of \((1 - x)/(1 - 2 x + 3 x^2)\)
• Smallest prime formed by concatenation \(n, n-1, \dots, n-k\)
• Conjectures associated with A087719
• Number of pairs of twin primes between \(n^2\) and \((n+1)^2\)
• Primality and primitive root property from divisibility \(n \mid (a(n-1) + 2^{n-2})\)
• Tug of war score between prime gap increases and decreases
• Product of composite numbers in triangular intervals
• \(\gcd(\mathrm{numerator}(H_n), n!)\)
• Recurrence \(a(n) = (a(n-1) + a(n-2)) \pmod n\)
• Coefficients in a simultaneous approximation to two polylogarithms
• Expansion of g.f. \((1+x)(3+x)/(1+6x^2+x^4)\)
• Conjectures associated with A100474
• Prime-th recurrence with reversal at each step

Pentanacci \(\pi\) sequence to $a(n) = \sum_{i=0}^n…

• Pentanacci \(\pi\) sequence
• Conjectures associated with A100800
• Conjectures associated with A101779
• Conjectures associated with A102371
• Floor of sum of \(\{n/k\}\)
• \(a(0) = 1\), \(a(n) = a(n-1)a(n-1) + 2\)
• Number of decompositions of \(2n+1\) into \(2p+q\), where \(p\) and \(q\) are both odd primes
• Fibonacci transform satisfying \(|a(n)| = F(n+1)\)
• \(a(n) = 3a(n-1) + a(n-2) - 3a(n-3)\)
• Smallest power with base>1 and exponent \(n\) without digit 0
• \(a(n) = \[x^{2n}\] \left(\frac{1 + x}{1 - x}\right)^n\)
• Number of zeros in ternary representation of \(2^n\)
• Array read by upward antidiagonals
• Conjectures associated with A105210
• Indicator sequence for 5 Fibonacci numbers with n digits
• Triangular matchstick numbers in the class of prime numbers
• Imaginary part of \(\prod_{k=0}^n (1 + k \cdot i)\), \(i = \sqrt{-1}\)
• Fibonacci-Collatz sequence
• Sum of squares of nonacci numbers
• \(a(n) = \sum_{i=0}^n \binom{2n-i}{n+i}\)

Riesel Problem to Number of refactorable numbers…

• Riesel Problem
• \(a(n) = 16n^2 + 1\)
• Digital sum of the Fermat number \(2^{2^n} + 1\)
• A108306: Expansion of \((3x+1)/(1-3x-3x^2)\)
• Numbers \(n\) such that \(\phi(n) = \phi(n + \phi(n))\)
• Numbers \(n\) such that the perfect deficiency of \(n\) is \(\le 10\).
• Numerator of \(\sum_{k=1}^n 2^k/k\).
• Numerator of \(\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)\)
• Conjectures associated with A109227
• Conjectures associated with A109671
• Conjectures associated with A109845
• Conjectures associated with A109905
• Conjectures associated with A109908
• Conjectures associated with A109909
• Number of symbols '*' and '^' to write the canonical prime factorization of n
• \(a(n) = \operatorname{lcm}\{1,2,\dots,n\}/\operatorname{denom}(H(n))\)
• Smallest \(m > 0\) such that there are no primes between \(nm\) and \(n(m+1)\) inclusive.
• Conjectures associated with A110854
• Integer part of \(\mathrm{prime}(n)/\pi(n)\)
• Number of refactorable numbers (A033950) \(\le 10^n\)

Sequence related to NOR… to $a(n) = \sum_{j=1}^{n} (3^j +…

• Sequence related to NOR bracketings
• A generalized Stern sequence
• Number of digits of n raised to the power of the sum of the digits of n
• Number of digits of n raised to the power of the digital root of n
• Smallest number \(m\) such that \(2^n - m\) and \(2^n + m\) are primes
• Expansion of g.f. \(-(1 - 48x^2 - 256x^3) / ((1 - 4x)(1 + 4x)(1 + 4x + 16x^2))\)
• Conjectures associated with A113252
• Square odd-indexed terms in the recurrence \(a(n) = 8a(n-1) - 8a(n-2) + 8a(n-3) - a(n-4)\)
• Conjectures associated with A113255
• Ascending descending base exponent transform of squares
• Ascending descending base exponent transform of factorials
• Ascending descending base exponent transform of \(2^n\)
• Number of prime powers \(q<=n\) such that also \(q+2\) is a prime power
• Difference between first odd semiprime \(> 2^n\) and \(2^n\)
• Largest odd divisor of \(a(n-1) + \textrm{prime}(n)\)
• Numerator of \(\zeta(4n)/\zeta(2n)^2\) (with \(a(0)=2\) instead of \(-2\))
• Each term is previous term plus floor of harmonic mean of two previous terms.
• Partial sums of \(\binom{2n}{n}^2\)
• \(a(n)\) = the number of values of \(k <= 10^n\) such that \(\sqrt{k(k+1)(k+2)(k+3)+1}\) is prime
• \(a(n) = \sum_{j=1}^{n} (3^j + (-2)^j)\)

Determinants of 2 X 2 matrices… to Denominators of $\sum_{k=1}^n…

• Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes
• Number of primes in \(n\)-th row of triangle \(k^2 - k + p_n\)
• Least \(k\) such that cyclotomic polynomial \(\Phi_k(n)\) is prime
• Sum of Fermat number and Mersenne number minus 1: \(2^{2^n} + 2^n - 1\)
• Least \(k \ge 1\) such that \(2 \cdot n^k - 1\) is prime
• Half-Fibonacci sequence
• Ratio of product of GCDs to product of factorials of floor divisions
• Odious primes minus evil primes among first \(n\) primes
• Recurrence involving LCM: \(a(n) = x(n+1)/x(n) - 2\)
• Number of Abelian cubes of length \(3n\) over an alphabet of size 3
• Factorial distance to nearest square
• Collatz step differences
• Representations as \(p + 2^x + 7 \cdot 2^y\) with \(p \equiv 5 \pmod 6\)
• Representations as \(p + 2^x + 11 \cdot 2^y\) with \(p \equiv 1 \pmod 6\)
• Smallest \(m\) such that \(n^3 + m^3 + 1\) is prime
• Number of ways to express \(n\) as sum of square, pentagonal, and hexagonal numbers
• Rowland-style prime-generating recurrence
• Chua's Euclidean prime sequence
• Smallest index \(k > n\) such that \((p_k+p_{k+1})/(p_n+p_{n+1})\) is an integer \(\ge 2\)
• Denominators of \(\sum_{k=1}^n \frac{1}{k 2^k}\)

Recurrence with fourth powers… to Number of primes of the form…

• Recurrence with fourth powers of binomial coefficients
• Central binomial sum \(a(n) = \sum_{k=0}^n (-4)^k \binom{n}{k}^2 \binom{n-k}{k}^2\)
• Central binomial sum \(a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k\)
• Difference of digit sums in base 3 and base 2
• Van Eck's sequence (OEIS A181391)
• Sum of fourth powers of Fibonacci-like binomial coefficients
• Product of two consecutive primes modulo the next prime
• Recurrence with bitwise XOR
• Number of odd primes between \(n^2\) and \((n+1)^2\) with \((n/p) = 1\)
• Coefficients of \(\prod_{k>0} (1 - x^k/k!)\)
• Number of ways to write \(n = x + y\) (\(0 < x < y < n\)) with \(6x \pm 1\) and \(6y \pm 1\) prime
• Normalized convolution of \(\binom{6k}{3k}\binom{3k}{k}\)
• Primorial squared plus one
• Sequence \(a(n) = n + \lfloor n r / t \rfloor + \lfloor n s / t \rfloor\)
• Linear recurrence \(a(n) = 5a(n-1) - 8a(n-2)\) with \(a(0)=0, a(1)=1\)
• Representations of \(n = p + q + (n \bmod 2)q\) with Jacobi symbol constraints
• Number of distinct sums of distinct proper divisors of \(n\)
• Size of smallest subset of \(\{1, 2, \dots, n\}\) with distinct subset sums
• Denominator of \(B_{n+1}(x) - B_{n+1}\)
• Number of primes of the form \(2^n \pm 2^k \pm 1\) with \(0 \le k < n\)

Primes of the form $3^n \pm 3^k… to Primes \(p < n\) with $n^2 - n +…

• Primes of the form \(3^n \pm 3^k \pm 1\)
• Lerch quotients of odd primes
• Position of \(n\)-th partial sum of the harmonic series when jointly ranked with \(\{\log(k+1)\}\)
• Expansion of \(\sum_{n \ge 0} \frac{(3n)!}{n!^3} \frac{x^{2n}}{(1-x)^{3n+1}}\)
• Least integer \(m > 1\) dividing no sum of two distinct primorials up to \(P_n\)
• Integrality and supercongruences of the factorial ratio \(\frac{(6n)! n!}{(3n)! (2n)!^2}\)
• Sequence \(\frac{(8n)! n!}{(4n)! (3n)! (2n)!}\)
• Balanced walks on a 4-dimensional cubic lattice with neighbor steps
• Partial sums of \((-1)^{k - \Omega(k)}\)
• Remainder of \(2^{n+2}\) modulo \(n\)
• Twin primes of the form \((3^n - k) 2^n \pm 1\)
• Number of \(0\)'s in the binary expansion of \(n^2\)
• Smallest deficient multiplier for non-deficiency
• Number of primes between \(n^3 - n\) and \(n^3\)
• Number of primes between \(n^2\) and \(n^2 + (\log_2 n)^2\)
• Recurrence \(a(n+1) = 2n(n+1)a(n) - n^4 a(n-1)\) and associated polynomials
• Smallest integer \(s > n\) such that \(1 + 2s + 3s^2 + \dots + n s^{n-1}\) is prime
• Partitions \(n = x + y\) with \(x^2 + xy + y^2\) prime
• Partitions \(2n + 1 = x + y\) with \(x^4 + y^4\) prime
• Primes \(p < n\) with \(n^2 - n + p\) and \(n^2 + n - p\) prime

Representations of \(n\) using… to Partitions \(n = p + q\) with…

• Representations of \(n\) using prime pairs with prime gaps of \(6\)
• Partitions of \(n\) into \(x + y\) with \((xy)^2 + 1\) prime
• Partitions of \(n\) into \(x + y\) with \((xy)^2 + xy + 1\) prime
• A double sum of five binomial factors
• Trajectory of \(64\) under the map \(n \to \text{A006368}(n)\)
• Existence of integers \(k\) with \(k^2 \operatorname{XOR} (k+1)^2 = (2n+1)^2\)
• Determinant of Legendre symbol matrix with factorial shifts
• Representation of sequence terms by harmonic numbers \(\lfloor \frac{1}{2H(n) - H(n^2+n-1) - \gamma} \rfloor\)
• Representations using Sophie Germain primes and twin prime pairs
• Eighth root of the generating function of an Apéry-like sequence
• Alternating sum of fourth powers of binomial coefficients
• Partitions of \(n\) into \(x + y\) with \(x(x+1)/2 + y^2\) prime
• Square root of the absolute value of the determinant of the prime sum indicator matrix
• Determinant of odd composite sum indicator matrix
• Determinant of prime sum/difference matrix
• Determinant of square/cube sum matrix
• Numbers \(n\) such that \(n^2 + \pi(n)\) is prime
• Undirected circular permutations with adjacent products minus one all prime
• Ways to write \(n = x + y + z\) with six linear and quadratic forms prime
• Partitions \(n = p + q\) with \(p\), \(3p - 10\), and \((p - 1)q - 1\) prime

Triples summing to \(n\) with… to Ordered ways to write $2n = p +…

• Triples summing to \(n\) with \(2m+1\) and \(2m^3+1\) prime
• Smallest \(n\)-th power equal to a sum of consecutive immediately preceding positive \(n\)-th powers
• Sum of two numbers with prime conditions
• Ways to write \(n = x + y\) with \(2^x + y(y+1)/2\) prime
• Sequence designed to show there are infinitely many primes congruent to \(1\) modulo \(4\)
• Representations with prime conditions
• Ways to write \(n = x + y\) with \(nx + y\) and \(ny - x\) both prime
• Least \(m > 0\) such that \(\{2^k - k : k = 1, \dots, m\}\) covers all residues modulo \(n\)
• Representations \(n = k^2 + m\) (\(0 < k^2 \le m\)) with \(\sigma(k^2) + \phi(m)\) prime
• Number of ways to write \(n = p + q\) (\(q > 0\)) with \(p\) and \((\phi(p)\phi(q))^4 + 1\) prime
• Number of primes \(p < n\) such that \(p \phi(n - p) - 1\) is also prime
• Values \(m = \sigma(k)\) (\(0 < m < 2n\)) with \(2n - 1 - m\) and \(2n - 1 + m\) prime
• Number of integers \(0 < k < n\) such that \(k \phi(n - k) + 1\) is a square
• Number of integers \(0 < k < n\) such that \((k + 1)^{\phi(n - k)} + k\) is prime
• Smallest \(x\) such that \(x \bmod \phi(x) = n\), or \(0\) if no such \(x\) exists
• Integers \(0 < k < n\) with \(p = k + \mathrm{prime}(n - k)\) and \(\mathrm{prime}(p) - p + 1\) prime
• Number of integers \(0 < k < n\) such that \(p = k + \phi(n - k)\) and \(2(n - p) + 1\) are both prime
• Number of \(0 < k < n-2\) with \(p = \phi(k) + \phi(n-k)/2 + 1\) and \(p_p - p \pm 1\) prime
• Prime quadruplets generated by \(p = 3\phi(k) + \phi(n-k) - 1\)
• Ordered ways to write \(2n = p + q\) with \(p, q\) and \(p_{p+2} + 2\) all prime

Number of \(0 < k < n/2\) such… to Difference between \(n \tau(n)\)…

• Number of \(0 < k < n/2\) such that \(\phi(k)\phi(n-k)\) is a square
• Number of parts in the symmetric representation of \(\sigma(n)\)
• Ordered ways to write \(n = k + m\) with \(p_k + 4\) and \(p_{p_m} + 4\) both prime
• Ways to write \(n = k + m\) with \(p_k^2 - 2\), \(p_m^2 - 2\), and \(p_{p_m}^2 - 2\) all prime
• Number of \(0 < k < n\) such that \(\pi(kn)\) is prime
• Number of primes \(p \le (n+1)/2\) with \(\lfloor\sqrt{n-p}\rfloor\) prime
• Pairs \(\{j, k\}\) with \(0 < j < k \le n\), \(k \equiv 1 \pmod j\), and \(\pi(jn) \mid \pi(kn)\)
• Equal prime counts in adjacent intervals \((kn, (k+1)n)\) and \(((k+1)n, (k+2)n)\)
• Number of positive integers \(k < n\) such that \(n^2 - \pi(k n)\) is prime
• Number of primes \(p < n\) with \(\text{prime}(p)^2 + (\text{prime}(n) - 1)^2\) prime
• Number of integers \(0 < k \le n\) such that \(\pi(\pi(k n))\) is a square
• Primitive roots of the form k² + 1
• Ways of writing \(n\) as an ordered sum of a triangular, square, and pentagonal number
• Largest smallest-base in a decomposition of \(n\) into at most four squares
• Least primitive prime divisor of the Franel numbers of order \(4\)
• Smallest \(k \ge 1\) such that the concatenation of \(k\) threes and \(\text{prime}(n)\) is prime
• Signed digit sums in bases \(b \ge 5\)
• Least index \(i\) for which \(\text{num}(\sigma_1(i)/i) - \text{den}(\sigma_1(i)/i) = n\)
• Sum of \(d \cdot \tau(d)\) over proper divisors \(d\) of \(n\)
• Difference between \(n \tau(n)\) and \(\sum_{d \mid n, d < n} d \tau(d)\)

Least \(m\) such that \(m + n\)… to Representations of \(n\) as $x^2…

• Least \(m\) such that \(m + n\) divides \(\mathrm{prime}(m) + \mathrm{prime}(n)\)
• Least \(m > 0\) coprime to \(n\) such that \(mn \mid C(m+n)\)
• Smallest prime factors of \(2^{2^n+2} + 3\)
• Smallest \(m \in \[1, n\]\) such that \(\binom{n}{m}\) is evil
• Odd part of the digital sum of \(3^n\) divided by the maximal power of \(3\)
• Floor of \(n^2\) divided by the sum of products of consecutive divisors of \(n\)
• Representations of \(n\) as the sum of two pentagonal and a generalized decagonal number
• Representations of \(n\) as a sum of generalized heptagonal, octagonal, and nonagonal numbers
• Partitions of \(n\) into distinct non-squarefree parts for \(n > 23\)
• Ways to write \(n\) as the sum of three elements of \(\{\lfloor T(x)/3 \rfloor : x \ge 1\}\)
• Asymptotics of an inner sum formula for \(a(n)\)
• Catalan numbers modulo \(6\)
• A GCD-Driven Sequence with Universal Jumps
• Prime values in recurrence \(a(n+1) = |a(n) - \gcd(a(n), 7n+6)|\)
• Primes \(p\) such that \(n - (p n' - 1)\) and \(n + (p n' - 1)\) are both prime
• Sums of four binary palindromes
• Refining Lagrange's four-square theorem with \((5x^2 + 7y^2 + 9z^2)yz\) a square
• Additive decompositions of \(\pi(T(n))\)
• Additive decompositions of \(\pi(\frac{n(n+1)}{2} + 1)\)
• Representations of \(n\) as \(x^2 + \phi(y^2) + \phi(z^2)\)

Sums of a cube, a square, and a… to Characterization of Carmichael…

• Sums of a cube, a square, and a positive triangular number
• Sums of a square and three weighted cubes
• Sums of a triangular number and three weighted cubes
• Recurrence for Rule 167 cellular automaton sequence
• Maximal run lengths in the second MSB of Tribonacci numbers \(T(n)\)
• Congruences \(a(n+k) \equiv a(n) \pmod k\) for hypergeometric values \({}_2F_1(n, -n; ; -1)\)
• The 1680-Conjecture
• The cup-of-coffee sequence
• Sum of four squares with square conditions
• Period of \(p\)-th powers modulo \(n\)
• Sum of a triangular number, a generalized pentagonal number, and a generalized heptagonal number
• Symmetry of digit sum differences \(\operatorname{sign}(S_5(k) - S_1(k))\)
• Step difference bound \(a(n+1) - a(n) \in \{1, 2\}\) in mass-redistribution cellular automata
• Sum of two squares and two central-binomial-type terms
• Sum of two squares, a power of 3, and a power of 5
• Maximality of \(k = 43\) with restricted digit counts in bases \(3 \le b < k\)
• The 2-4-6-8 Conjecture
• Vanishing of bases with \(b^n \equiv -1 \pmod{2n+1}\) for absolute Euler pseudoprimes
• Four-square conjecture with powers of 2, 3, and 5
• Characterization of Carmichael numbers via squarefree denominators \(a(n)\)

Nonnegativity of the Dirichlet… to A cubic generating function…

• Nonnegativity of the Dirichlet square root of A046644
• Pronic indices for odd coefficients of \(\sum_{n \ge 0} x^n \frac{(1+x^n)^n}{(1+x^{n+1})^{n+1}}\)
• Pronic indices for odd coefficients of \(\sum_{n \ge 0} x^n \frac{(1+x^n)^n}{(1-x^{n+1})^{n+1}}\)
• Tripling until the seed's digits reappear
• Convergence of fraction numerators \(a(n)/b(n)\) to \(e\)
• Bounds on fractional parts \(n r^2 - a(n)\) for \(r = (2+\sqrt{5})/2\)
• Numerator of a sum involving binomial coefficients
• Factor bounds for Fermat numbers
• A factorial ratio analogous to the Catalan numbers
• Primality of continued fraction denominators \(a(n)\) for \(n \ge 3\)
• Realization of primes \(p \equiv \pm 1 \pmod{10}\) by continued fraction denominators
• A generating function with powers two and three
• A generating function with powers two and four
• A generating function with powers two and four
• A quadratic generating function with sign substitution
• A generating function with signed square terms
• The generating function \(A(x)=1+3xA(x)^2-2xA(-x)^2\)
• Integrality of the factorial ratio \(\frac{(12n+6)! (6n+9)!}{108 (4n+2)! (2n+3)! ((6n+5)!)^2}\)
• Unique realization of odd primes \(p \notin \{3, 5\}\) by continued fraction values
• A cubic generating function with ternary substitution

A logarithmic derivative with… to Decimal concatenation of a…

• A logarithmic derivative with prime-divisible coefficients
• Periodicity of \(k\)-th prime factors in coupled nonlinear recurrence \(a(n)\)
• Lengths of the 2-dense sublists of divisors of \(n\)
• A differential generating function
• Maximum number of terms in a 2-dense sublist of divisors of \(n\)
• Decimal concatenation of a cofactor and the largest prime factor

No article: 2 files here state no research or textbook statement: \(a(n) = n\) if \(n\) is prime, otherwise \(a(n) = 1\), Sloping binary numbers: read array of binary numbers (right-justified) along diagonals of slope \(-1\).

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