cp's OEIS Frontend

Previous name "Largest unitary square divisor of n!" was incorrect. See A374988 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

This sequence is multiplicative. - Mitch Harris, Apr 19 2005

The sequence enumerates the solutions of a system of polynomials equations modulo n, hence is multiplicative by the Chinese Remainder Theorem. The middle entry of the 3 X 3 is zero modulo n. - Michael Somos, Apr 30 2005

Equivalently, numbers k whose squarefree part, A007913(k), is smaller than their square part, A008833(k). - Peter Munn, Mar 26 2021

Complementary to A048803 which can be defined as squarefree factorials.

Partial products of A008833. - Ray Chandler, Nov 29 2004

The equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(a(n)), where M and N are integers. Moreover, it is solvable in numbers of the form N + M*sqrt(l), where l has the form l = A007913(k^4 - 4*k*m^3), where k,|m| >= 1 (without restrictions k,|m| <= 5*l). But in this more general case there could be unknown numbers l having this form; this circumstance does not allow construction of the full sequence of such l. Therefore we restrict ourselves by the condition k,|m| <= 5*l. Note that testing l with respect to this condition is rather simple by sorting all values of k,|m| <= l. One can prove that, at least, if the Fermat numbers (A000215) are squarefree, then the sequence is infinite. Conjecture (necessity of the form of l): If the equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(l) with integer N,M, then there exist positive integers k,m such that l = A007913(k^4 - 4*k*m^3).

Oex divisors d of an integer n are defined in A186443: those divisors d which are either 1 or numbers such that d^k || n (the highest power of d dividing n) has odd exponent k.

A positive number is called an oex prime if it has only two oex divisors; since every n >= 2 has at least two oex divisors, 1 and n, an oex prime q has only oex divisors 1 and q. A000430 is the sequence of oex primes q, i.e., A186643(q) = 2 iff q is an entry in A000430.

A unique factorization, called an oex prime power factorization, of integers n is introduced as follows: each factor p^e in the conventional prime power factorization n = Product(p^e) is written as (p^2)^(e/2) if e is even, and as (p^2)^floor(e/2)*p if e is odd. This represents n as a product of oex primes of the type q=p^2, with unconstrained exponents e/2, and of oex primes of the type q=p with exponents 0 or 1. (This is similar to splitting n into its squarefree part A007913(n) times A008833(n), followed by an ordinary prime factorization in both parts separately.)

Let n = q_1^a_1*q_2^a_2*... and m = q_1^b_1*q_2^b_2*..., a_i,b_i >= 0 be the oex prime power factorizations of n and m. Define the oex GCD of n and m as [n,m] = q_1^min(a_1,b_1) * q_2^min(a_2,b_2) * .... Then a(n) = Sum_{m=1..n, [m,n]=1} 1, the oex analog of the Euler-phi function.

The complementary sequence, squarefree part of k is one greater than the square part of k, is A069187.

a(n) are the partial products of A219964(n).

a(n) divides n!, n!/a(n) = 1, 1, 1, 1, 2, 2, 4, 4, 8, 72, 144, 144, 192...

The swinging factorial (A056040) divides a(n), a(n)/n$ = 1, 1, 1, 1, 2,...

The primorial of n (A034386) divides a(n), a(n)/n# = 1, 1, 1, 1, 2, 2, 6,..

If p^k is the largest power of a prime dividing a(n) then k is 2^n for some n >= 0.

a(n) / A055773(n) is the largest square dividing a(n), a(n) / A055773(n) = A008833(a(n)).

If the numbers > 1 are replaced by 1 one obtains the corresponding characteristic triangle. a(n,n) = n. a(n,1) = sqrt(n) iff n is a square.

The number of nonzero entries in row n is A000188(n).

For n and m with gcd(n,m) = 1 the nonzero entries are precisely a(N^2,M^2) = N*M, with integers N, M satisfying gcd(N,M) = 1 , 1 <= M <= N. - Wolfdieter Lang, Apr 26 2013