Practical numbers
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number. - Hal M. Switkay, Jan 28 2023
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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).
Prime-th recurrence with reversal at each step
Starting at a positive value other than a(0) = 1, does this sequence ever go into a loop? The positivity hypothesis is required because the source recurrence uses the one-based prime index p₁ = 2; the x = 0 branch above is only an artifact of making aStartAt total on ℕ.
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Primes of the form 2^n + 2^i + 1
Conjecture (A81091): There are infinite primes of the form 2^n + 2^i + 1, with 0 < i < n.
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Primitive roots of the form k² + 1
Zhi-Wei Sun's Conjecture (A239957): Every prime p has a primitive root 0 < g < p of the form k^2 + 1, where k is an integer.
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Product of two consecutive primes modulo the next prime
Conjecture: For x > 10^9, the most frequent value in a(n), n=1… x, has form 120k.
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Recurrence a(n) = (a(n-1) + a(n-2)) pmod n
All numbers appear infinitely often, i.e., for every number k ≥ 0 and every frequency f > 0 there is an index i such that a(i) = k is the f-th occurrence of k in the sequence. - _Klaus Brockhaus_, Aug 29 2006
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Recurrence involving LCM: a(n) = x(n+1)/x(n) - 2
Conjecture: For prime p such that p-2 is not a prime, a(p-1) = p. - _Bill McEachen_, Sep 26 2025
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Recurrence with bitwise XOR
Conjecture: the sequence contains 8 zeros.
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Representations as p + 2^x + 11 · 2^y with p ≡ 1 pmod 6
On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n) = 0 if and only if n < 16 or n ∈ 18, 21, 24, 51, 84, 1011, 59586; in other words, except for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 30 can be written as the sum of a prime congruent to 1 bmod 6, a positive power of 2 and…
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Representations with prime conditions
Zhi-Wei Sun's Conjecture (A232174): Any integer n > 1 can be written as x + y with x, y > 0 such that both x + ny and x^2 + ny^2 are prime.
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Riesel Problem
It is conjectured that the integer k = 509203 is the smallest Riesel number, that is, the first n such that a(n) = -1 is 254602.
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Rowland-style prime-generating recurrence
Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).
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Smallest composite c such that textrmprimorial(n) + c is prime
Conjecture: liminf_n → ∞ a(n)/p_n+1^2 = 1 < limsup_n → ∞ a(n)/p_n+1^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015
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Smallest factorial containing exactly n 6's
It is conjectured that a(24) = 0 since no factorial less than 10000 contained just 24 sixes.
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Smallest index k > n such that (p_k+p_k+1)/(p_n+p_n+1) is an integer ≥ 2
Conjecture: the sequence is infinite, that is, for every n ≥ 1 there is some k > n with S(n) | S(k), so that a(n) is defined.
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Smallest m > 0 such that n · 2^m + 1 is prime
There is a conjecture that the first zero is n = 65536 = 2^16 (which is equivalent to the statement that 2^2^k + 1 is composite for k > 4). - _T. D. Noe_, Feb 25 2011
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Smallest m > 0 such that there are no primes between nm and n(m+1) inclusive.
Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n.
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Smallest m such that n^3 + m^3 + 1 is prime
Conjecture 2: For any k ≥ 3, there are infinitely many primes of the form n^k + m^k + 1 for n, m ≥ 1. - _Ulrich Krug_, 2009
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Smallest number k such that kn + 1 is prime
Conjecture: for every n > 1 there exists a number k < n such that nk + 1 is a prime.
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Smallest number m such that 2^n - m and 2^n + m are primes
Conjecture: a(n) = O(n^3). The source defines a(n) as the least m with 2^n - m and 2^n + m prime, so it implicitly asserts that such an m exists. Since a n = 0 when no such m exists, the existence of a prime pair is stated explicitly for all sufficiently large n.
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Smallest palindrome with exactly n divisors
There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number.
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Smallest prime ≡ 1 pmod n
"Conjecture: a(n) < n^2 for n > 1. - _Thomas Ordowski_, Dec 19 2016"
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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 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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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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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 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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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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Tug of war score between prime gap increases and decreases
Is the score a(n) > 0 for some n > 250000?
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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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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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