A binomial coefficient sum
Let b(n) = a(2n-1). Then the supercongruence b(n p^k) ≡ b(n p^k-1) pmodp^3k holds for positive integers n and k and all primes p ≥ 5. - Zhi-Wei Sun, Nov 16 2019
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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).
A binomial coefficient summation
Let b(n) = a(2n-1). Then the supercongruence b(n p^k) ≡ b(n p^k-1) pmodp^3k holds for positive integers n and k and all primes p ≥ 5. - Zhi-Wei Sun, Nov 16 2019
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A GCD-Driven Sequence with Universal Jumps
Does every positive integer occur as a difference in this sequence?
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a(0) = 1, a(n) = a(n-1)a(n-1) + 2
Prime for a(1) = 3, a(2) = 11, a(4) = 15131; semiprime for a(3) = 123 = 3 41, a(5) = 228947163 = 3 76315721. a(6), added by Jonathan Vos Post, has 4 prime factors. a(7) = 41 811^2 106693969 317171188688357726699 8272236925540996054440172449761. When is the next prime in the sequence?
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a(n) = (smallest prime > n^2) - n^2
Conjecture: a(n) ≤ 1 + φ(n) for n > 0. This improves on Oppermann's conjecture, which says a(n) < n. - Thomas Ordowski, Dec 17 2014
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a(n) = 2^(2^n)
I conjecture that a(n) ; n>1 are the numbers such that n^4-1 divides 2^n-1, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015 This formalizes the reverse direction.
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a(n) = 3a(n-1) + a(n-2) - 3a(n-3)
The current sequence contains primes, including 3, 5, 41, 21523361. Is there an (a, b, c) weighted tribonacci sequence with a, b, c relatively prime which is prime-free?
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a(n) = lcm1,2,…,n/denom(H(n))
It is conjectured that every odd number occurs in this sequence.
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a(n) = the number of values of k <= 10^n such that √(k(k+1)(k+2)(k+3)+1) is prime
Conjecture: a(n)/A006880(n) → 1.77... where A006880(n) is the number of primes ≤ 10^n.
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a(n) = Σ_j=1^n (3^j + (-2)^j)
First primes are a(11) = 264353 and a(17) = 193622861. Additional primes: a(71), a(91), a(431). What is the next prime?
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a(n) = Σ_k=0^n C(2k, k)^3
Conjecture 1 (Peter Bala, 2024): If prime p is in A003625 then a(p^2) ≡ 8 + p^2 pmodp^3.
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a(n) is the integer whose decimal digits are the first n+1 decimal digits of π
Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than 10^-9.
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Alternating sum of signs of powers of 3/2
For n large enough, does a(n) > √(n) always hold?
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Apéry numbers
For each n = 1, 2, 3, … the polynomial a_n(x) = Σ_k=0^n C(n, k)^2 C(n+k, k) x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
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Array read by upward antidiagonals
A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers 2n+1 and 2n+3 for n > 0, at least one will be the product of 2 primes (not necessarily distinct).
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Ascending descending base exponent transform of 2^n
The first prime terms in this (always odd) sequence are a(1) = 3, a(3) = 41, and a(4) = 593. What is the next prime? The OEIS comment currently says a(5) = 543, but this conflicts with its defining formula, b-file, and examples: the actual index-five term is the composite number 135457.
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Ascending descending base exponent transform of factorials
Is there a nontrivial power after a(4) = 5^3?
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Ascending descending base exponent transform of squares
The smallest prime in this sequence is a(2) = 5. What is the next prime?
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Binary representation of primes that divide a number, in decimal
Starting at any n and iterating the map n ↦ a(n), we will always reach 0. - _Antti Karttunen_, Jun 18,20 2017
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Catalan-Mersenne numbers
Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.
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Central binomial sum a(n) = Σ_k=0^n (-4)^k C(n, k)^2 C(n-k, k)^2
If p is a prime with p ≡ 1, 9 pmod20 and p = x^2 + 5y^2 with x, y integers, then Σ_k=0^p-1 a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 01 2010
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Central binomial sum a(n) = Σ_k=0^n C(n, k)^2 C(n-k, k)^2 (-16)^k
If p is a prime with (p/7) = 1 and p = x^2 + 7y^2 with x, y integers, then Σ_k=0^p-1 (-1)^k a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 17 2010
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Central trinomial coefficients
An integer n > 3 is prime if and only if a(n) ≡ 1 pmodn^2. We have verified this for n up to 8 · 10^5, and proved that a(p) ≡ 1 pmodp^2 for any prime p > 3 (cf. A277640). - Zhi-Wei Sun, Nov 30 2016
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Chua's Euclidean prime sequence
Does Chua's sequence contain every prime?
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Coefficients of Π_k>0 (1 - x^k/k!)
The coefficients c(n) of A(x)^2 = (Σ_n ≥ 0 a(n) x^n)^2 differ in sign from c(n-1) if and only if n is a triangular number. - _Peter Bala_, Mar 17 2022
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Collatz step differences
Conjecture 1: More than half of the terms are 0. - _Ya-Ping Lu_, May 04 2024
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Concatenation of the next n numbers
"The second term is a prime. When is the next prime, if there is another? - _N. J. A. Sloane_, Dec 16 2016"
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Conjectures associated with A038552
All terms of A038552 are congruent to 19 pmod24.
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Conjectures associated with A063880
All members of the sequence satisfy n ≡ 108 pmod216.
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Conjectures associated with A067720
For members of the sequence other than 8, we have k + 1 is prime.
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Conjectures associated with A100474
After a(2) = 5, is there another prime?
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Conjectures associated with A100800
A100800 Conjecture: No term is zero.
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Conjectures associated with A101779
It is conjectured k always exists.
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Conjectures associated with A105210
Cormier and Selfridge found 5 starting values for which the sequences appear to not merge. The sequences were checked up to 10^8.
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Conjectures associated with A109227
Conjecture: a(2) and a(121) are primes. Are there any more?
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Conjectures associated with A109671
Does the sequence contain every positive integer (cf. A169741)?
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Conjectures associated with A109845
Conjecture: There are infinitely many primes in this sequence.
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Conjectures associated with A109905
a(n) = 0 for n = 1, 6, 30 and 54. Are there any others?
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Conjectures associated with A109908
Conjecture: a(n) > 0 for n > 3.
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Conjectures associated with A109909
Conjecture: a(n) > 0 for n > 3.
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Conjectures associated with A110854
Do the absolute values cover A004275? A004275 is 1 together with the nonnegative even numbers. The conjecture asks whether every member of A004275 occurs as |a(n)| for some term of the sequence.
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Cuban Primes
This sequence is believed to be infinite.
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Decimal encoding of the prime factorization of n
"a(31) = a(177147) = 311. Is there any solution to a(n) = n? - _Franklin T. Adams-Watters_, Dec 18 2006"
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Denominator of sum of reciprocals of divisors
If a(n) is in A005153, then n is in A005153. - Jaycob Coleman, Sep 27 2014 We require 0 < n because a(0) = 1 is in A005153 (practical numbers), but 0 is not.
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Denominator of Σ_k=1^n k^μ(k)
Conjecture: a(n) = primorial(n) for infinitely many n.
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Denominators of coefficients in Stirling's expansion for log(Γ(z))
Conjecture I: if n > 2, then a(A005382(n))/12 is prime, where A005382 is the sequence of primes p such that 2p-1 is also prime. Since A005382(1) = 2, A005382(2) = 3 and A005382(3) = 7, this says that a(p)/12 is prime for every prime p > 3 such that 2p-1 is also prime.
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Determinant of Hankel matrix of the first 2n-1 prime numbers
"I conjecture that a(4) is the only zero. - _Jon Perry_, Mar 22 2004" Stated as a biconditional: the claim that a(4) is the only zero asserts both that a(4) = 0 and that no other index vanishes. A bare implication a n = 0 → n = 4 would be satisfied vacuously by a sequence with no zero at all.
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Determinant of matrix with entries indicating primality of i^2 + j^2
Conjecture: a(n) = 0 for no n > 28. - _Zhi-Wei Sun_, Aug 26 2013
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Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes
This suggests the ratio is approaching a limit close to 0.87. Formalized as: The sequence of ratios P(N)/Neg(N) converges to a limit L, and L is in the interval (0.8, 0.9).
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Difference between first odd semiprime > 2^n and 2^n
In this powers of 2 sequence, does 1 occur infinitely often?
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Difference of digit sums in base 3 and base 2
"This sequence is positive on average, since 1/log(3) > 1/log(4). Do all integers appear infinitely often?" - Charles R Greathouse IV, Feb 07 2013
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Digital sum of the Fermat number 2^2^n + 1
a(0), a(1), a(5), a(6), a(7) and a(11) are primes. Are there any more?
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Divisibility of 2^n + 1 by n
All members of the sequence A56777 come from prime quadruples.
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Euclid-Mullin sequence
"Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski and later by Shanks, [Wagstaff93] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored.
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Expansion of (1 - x)/(1 - 2 x + 3 x^2)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017
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Factorial distance to nearest square
This sequence suggests that the distance between a factorial and the closest power is tightly bounded.
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Four-square conjecture with powers of 2, 3, and 5
Zhi-Wei Sun's Four-Square Conjecture (A308734): Any integer n > 1 can be written as (2^a · 3^b)^2 + (2^c · 5^d)^2 + x^2 + y^2 for nonnegative integers a, b, c, d, x, y.
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gcd(numerator(H_n), n!)
Conjecture: Every odd prime occurs as a term in the sequence.
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Half-Fibonacci sequence
Conjecture (1): The natural density of even terms in the sequence is 1/2.
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Home primes (OEIS A037274)
Every integer at least two reaches a home prime.
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Integer part of area of a regular polygon with n sides each of length 1
"Usually (perhaps always?) ⌊ n^2 / (4π) - π / 12 ⌋ for a polygon of circumference n. Note that the area of a circle with circumference C is C^2 / (4π)."
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Integer part of prime(n)/π(n)
Conjecture: As n → ∞, there are infinitely many n's such that a(n) is greater than a(n+1).
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Largest odd divisor of a(n-1) + textrmprime(n)
Is a(33900) the last term equal to 1?
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Least k > 0 such that (k+1)(k+2)⋯(k+n) + 1 is prime
(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2, which is never prime. Hence a(4) = 0. Conjecture: a(n) = 0 if and only if n = 4.
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Least k ≥ 1 such that 2 · n^k - 1 is prime
Is a(n) defined for all n ≥ 2? That is, does there exist k > 0 such that 2 · n^k - 1 is prime?
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Least k such that cyclotomic polynomial Φ_k(n) is prime
Is a(n) defined for all n ≥ 1? That is, for every n ≥ 1, does there exist k > 0 such that |Φ_k(n)| is prime?
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Least m such that φ(m) = n!
Conjecture: unless n! + 1 is prime (i.e., n ∈ A002981), a(n) = p q where p is the least prime > √(n!) such that (p - 1) | n! and q = n!/p - 1 + 1 is prime. - M. F.
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Least positive multiple of n in base 10 with digits 0 and 1
It is known that a(10^k - 1) = (10^9k - 1) / 9 for all k. Is a(n) < a(10^k - 1) for all n < 10^k - 1? - David Radcliffe, Aug 01 2025
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Least prime ≥ n
According to the "k-tuple" conjecture, a(n) is the initial term of the lexicographically earliest increasing arithmetic progression of n primes; the corresponding common differences are given by A061558.
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Maximum exponent in the prime factorization of n
Are there composite numbers n > 4 such that n ≡ a(n) pmodφ(n)? - Thomas Ordowski, Dec 02 2019 This question is equivalent to Lehmer's totient problem LehmerTotient.lehmer_totient; a positive answer here falsifies the universal statement asked about in Erdos828.erdos_828.variants.lehmer_conjecture.
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Multiplicative order of 2 mod 2n+1
If p is an odd prime then a((p^3-1)/2) = p · a((p^2-1)/2). Because otherwise a((p^3-1)/2) < p · a((p^2-1)/2) iff a((p^3-1)/2) = a((p-1)/2) for a prime p. Equivalently p^3 divides 2^p-1-1, but no such prime p is known. - Thomas Ordowski, Feb 10 2014
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No powers as partition numbers
There are no partition numbers a(k) of the form x^m, with x,m integers >1. See comment by Zhi-Wei Sun (Dec 02 2013).
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Number of Abelian cubes of length 3n over an alphabet of size 3
Conjecture (Peter Bala, 2022): The supercongruences a(n · p^k) ≡ a(n · p^k-1) pmodp^3k hold for the integer-indexed extension a(n) for all n ∈ ℤ ∖ 0, primes p ≥ 5, and k ≥ 1.
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Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes
Conjecture: all items for n ≥ 4 are greater than or equal to 1. This is a stronger conjecture than the Goldbach conjecture.
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Number of different products of subsets of 1, 2, …, n
Conjecture: let p ≤ n be prime. If m and p^a m are two such products, then so is p^k m for all 0 < k < a. - Yan Sheng Ang, Feb 13 2020
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Number of digits of n raised to the power of the sum of the digits of n
n=1 and 32 are two fixed points. Are there any others?
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Number of odd primes between n^2 and (n+1)^2 with (n/p) = 1
Conjecture: a(n) > 0 for all n > 0. - _Zhi-Wei Sun_, Dec 29 2012
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Number of pairs of twin primes between n^2 and (n+1)^2
It is conjectured that a(n)>0 for all n>122. Proving this would also prove Legendre's conjecture that there is a prime between n^2 and (n+1)^2. - _T. D. Noe_, Feb 28 2007
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Number of prime powers q<=n such that also q+2 is a prime power
(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6).
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Number of prime powers strictly between n-th prime and (n+1)-th prime
It is conjectured that a(n) ≤ 2 for all n.
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Number of primes < n^2
Conjecture: all the numbers Σ_i=j^k 1/a(i) with 1 < j ≤ k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
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Number of primes < n^3
Conjecture (i): for any integer k > 2, the sequence π(n^k)/n^k (n = 2, 3, …) is strictly decreasing, where π(x) denotes the number of primes not exceeding x. - Zhi-Wei Sun, Oct 17 2015
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Number of primes in n-th row of triangle k^2 - k + p_n
Conjecture: a(n) < n for n > 13.
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Number of primes p such that 2^n ≤ p ≤ 2^n + prime(n)
For any n > 0, is there always at least one prime p such that 2^n ≤ p ≤ 2^n + prime(n)? (checked up to n = 250).
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Number of primes p such that n^n ≤ p ≤ n^n + n^2
Question: for any n > 0, is there at least one prime p such that n^n ≤ p ≤ n^n + n^2? In this case, that would be stronger than the Schinzel conjecture: "for m > 1 there's at least one prime p such that m ≤ p ≤ m + log(m)^2" since n^2 < log(n^n)^2 = n^2 log(n)^2.
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Number of refactorable numbers (A033950) ≤ 10^n
Colton's conjecture [Co99] as stated by Zelinsky [Ze02]: for every n, the number of refactorable numbers ≤ n is at least half the number of primes ≤ n, i.e. π(n) ≤ 2 T(n).
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Number of squares bmod n
n^2 ≡ 1 pmoda(n)(a(n)-1) if and only if n is an odd prime. - Thomas Ordowski, Jun 08 2017
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Number of symbols '*' and '^' to write the canonical prime factorization of n
It is conjectured that 1,2,3,4,5,6,7,9,11 are the only positive integers which cannot be represented as the sum of two elements of indices n such that a(n) = 1.
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Number of times n^2 + s^2 is prime for positive integers s < n
Conjecture: a(n) > 0 for all n > 1.
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Number of ways to express n as sum of square, pentagonal, and hexagonal numbers
In April 2009, _Zhi-Wei Sun_ conjectured that a(n) > 0 for every n = 0, 1, 2, 3, ….
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Number of zeros in ternary representation of 2^n
Conjecture from N. J. A. Sloane: a(n) > 0 for n > 15.
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Numbers n such that n^2 + π(n) is prime
Conjecture: the sequence A228828 is infinite.
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Numbers n such that the perfect deficiency of n is ≤ 10.
Is 1155 the last odd number in this sequence? (1155 is the 59th term starting from 1, corresponding to a(58) = 1155).
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Numbers n such that φ(n) = φ(n + φ(n))
Conjecture: Except for the first term all terms are even.
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Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)
"Conjecture: 1/det(M) is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015
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Numerator of a sum involving binomial coefficients
We conjecture that u(p-1) == 0 (mod p^4) for all primes p, with a finite number of exceptions that depend on m.
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Numerator of ζ(4n)/ζ(2n)^2 (with a(0)=2 instead of -2)
Conjecture: if an integer n > 1 is odd, then ζ(2n)/ζ(n)^2 is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022
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Numerator of Σ_k=1^n 2^k/k.
Conjecture: for n > 3, textrmnumerator(-2/n + Σ_k=1^n 2^k/k) == 0 (textrmmod n^2) if and only if n is prime.
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Odious primes minus evil primes among first n primes
Shevelev conjectures that a(n) ≥ 0 for n > 3.
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Partial sums of C(2n, n)^2
Conjecture: For any positive integer n, the polynomials Sum_k=0^n binomial(2k,k)^2x^k and Sum_k=0^n binomial(2k,k)^2x^k/(k+1) are irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 23 2013
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