Smallest prime ≡ 1 pmod n
"Conjecture: a(n) < n^2 for n > 1. - _Thomas Ordowski_, Dec 19 2016"
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Open conjectures from the OEIS
Conjectures stated in OEIS entries — that a sequence is infinite, that a formula holds for all n, that some search never ends — formalised in Lean by Formal Conjectures. Many can be tested by computing more terms, and a counterexample is a checkable certificate.
Source: The On-Line Encyclopedia of Integer Sequences. Licence: Lean statements from Formal Conjectures (Apache 2.0).
Smallest prime formed by concatenation n, n-1, …, n-k
Conjecture: There are infinitely many composite numbers n such that a(n) is nonzero.
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Smallest r such that (concatenation of n, r times) · 10 + 1 is prime
What is the smallest integer m > 1 such that a(10^m) is nonzero? - _Farideh Firoozbakht_, Jan 07 2015
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Smallest x such that x bmod φ(x) = n, or 0 if no such x exists
Conjecture: a(n) > 0 for all n.
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Smallest x such that σ(x) bmod x = n
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 σ_1(x) bmod x = 5.
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Square root of the absolute value of the determinant of the prime sum indicator matrix
We conjecture that a(n) > 0 for all n > 15.
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Sum of d · τ(d) over proper divisors d of n
Conjecture: 21 is the only positive integer n such that a(n) = n.
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Sum of Fermat number and Mersenne number minus 1: 2^2^n + 2^n - 1
The first 5 entries are primes. Are there infinitely many primes in this sequence?
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Sum of four squares with square conditions
Zhi-Wei Sun's Conjecture (A281976): Any integer n ≥ 0 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers and z ≤ w, such that both x and x + 24y are squares.
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Sum of next n primes
The only positive integer n such that a(n) is a perfect square is n=38. - Carlos Eduardo Olivieri, Mar 09 2015
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Sum of squares of divisors of n
Conjecture: For each k = 2,3,..., all the rational numbers σ_k(n)/n^k = Σ_d|n 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015
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Sum of squares of nonacci numbers
Primes in this sequence include: a(8) = 2, which is next?
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Sum of two numbers with prime conditions
The conjecture for sequence A231201: for any n > 1, there exist x, y > 0 such that n = x + y and 2^x + y is prime.
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Sum of two squares, a power of 3, and a power of 5
Zhi-Wei Sun's Conjecture (A303656): Any integer n > 1 can be written as the sum of two squares, a power of 3, and a power of 5.
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Sums of a cube, a square, and a positive triangular number
Conjecture: a(n) > 0 for all n > 0.
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Sums of a square and three weighted cubes
Conjecture (i): For any m ∈ 3, 4, 5, 6 and n ≥ 0, there are nonnegative integers w, x, y, z such that n = w^2 + x^3 + 2y^3 + mz^3.
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Sums of a triangular number and three weighted cubes
Conjecture (i): Any positive integer can be written as w(w+1)/2 + x^3 + by^3 + cz^3 with w > 0 and x, y, z ≥ 0, provided that (b, c) ∈ (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,20), (2,21), (2,34), (3,3), (3,4), (3,5), (3,6), (4,10).
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The 1680-Conjecture
Zhi-Wei Sun's 1680-Conjecture (A280831): Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x^4 + 1680 y^3 z is a square.
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The cup-of-coffee sequence
Conjectured to be a permutation of the positive integers.
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The prime numbers
Conjecture from Thomas Ordowski (2023): log log a(n+1) - log log a(n) < 1/n for n > 0.
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Trajectory of 103 under the Reverse and Add! operation in base 3
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.
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Trajectory of 64 under the map n → A006368(n)
It is conjectured that this trajectory does not close on itself.
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Triangular matchstick numbers in the class of prime numbers
Terms are squares at only(?) three values of n = 3, 6, 4072: corresponding terms are 6^2, 13^2, and 15735^2.
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Triples summing to n with 2m+1 and 2m^3+1 prime
Conjecture (i): a(n) > 0 for all n > 2.
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Tripling until the seed's digits reappear
OEIS A331631 states: The sequence is conjectured to be a permutation of the positive integers (see the Crossrefs section).
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Tug of war score between prime gap increases and decreases
Is the score a(n) > 0 for some n > 250000?
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Twin primes of the form (3^n - k) 2^n ± 1
Conjecture: There is always one such k for each n > 0 such that (3^n - k) 2^n - 1 and (3^n - k) 2^n + 1 are a twin prime pair.
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Undirected circular permutations with adjacent products minus one all prime
Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.
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Value of n-th cyclotomic polynomial at n
a(28341) is divisible by 283411^2. What is the next n such that a(n) is not squarefree?
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Values m = σ(k) (0 < m < 2n) with 2n - 1 - m and 2n - 1 + m prime
Conjecture (i): a(n) > 0 for all n > 3.
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Van Eck's sequence (OEIS A181391)
OEIS A181391 conjectures: "every number eventually appears."
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Ways of writing n as an ordered sum of a triangular, square, and pentagonal number
It is conjectured that a(n) is always positive (Conjecture 1.1 of Sun (2009)).
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Ways to write n = k + m with p_k^2 - 2, p_m^2 - 2, and p_p_m^2 - 2 all prime
Conjecture: a(n) > 0 for all n > 1.
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Ways to write n = x + y + z with six linear and quadratic forms prime
Conjecture: a(n) > 0 for all n > 5.
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Ways to write n = x + y with 2^x + y(y+1)/2 prime
Conjecture: a(n) > 0 for all n > 1.
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Ways to write n = x + y with nx + y and ny - x both prime
Conjecture (i), part 1: a(n) > 0 for all n > 2.
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Ways to write n as the sum of three elements of ⌊ T(x)/3 ⌋ : x ≥ 1
Conjecture: For any positive integer m, every nonnegative integer n can be written as ⌊ T(x)/m ⌋ + ⌊ T(y)/m ⌋ + ⌊ T(z)/m ⌋ with x, y, z nonnegative integers.
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Wolstenholme numbers
Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). - Amiram Eldar and Thomas Ordowski, Jul 28 2019
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