A366807 -id:A366807 - cp's OEIS frontend

1, 4, 9, 25, 12, 49, 20, 121, 169, 28, 45, 289, 361, 63, 44, 529, 52, 841, 60, 961, 99, 68, 175, 1369, 76, 117, 1681, 84, 1849, 92, 2209, 153, 2809, 275, 171, 116, 3481, 3721, 124, 325, 132, 4489, 207, 140, 5041, 5329, 148, 539, 156, 6241, 164, 6889, 425, 172

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across squarefree numbers A005117.
a(1) = 1 by definition. 1 is the empty product and has no least prime factor.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k > 1 is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k.
Plainly, k is the first term in the sequence k*R_k, because 1 is the first term in R_k. Hence a(n) is the second term in k*R_k for n > 1, since lpf(k) is the second term in R_k.

			Let b(n) = A005117(n).
a(2) = 4 = b(2)*lpf(b(2)) = 2*lpf(2) = 2*2. In {2*A000079}, 4 is the second term.
a(5) = 12 = b(5)*lpf(b(5)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term..
a(11) = 45 = b(11)*lpf(b(11)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001248, A005117, A020639, A065642, A073481, A120944, A285109, A364996, A366807, A366825.

nn = 120; s = Select[Range[nn], SquareFreeQ];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

apply(x->(if (x==1,1, x*vecmin(factor(x)[,1]))), select(issquarefree, [1..150])) \\ Michel Marcus, Dec 17 2023

from math import isqrt
from sympy import mobius, primefactors
def A366786(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return (m:=bisection(f))*min(primefactors(m),default=1) # Chai Wah Wu, Aug 31 2024

a(n) = A065642(A005117(n)), n > 1.
a(n) = A285109(A005117(n)).
a(n) = A020639(A005117(n))*A005117(n).
For prime p, a(p) = p^2.
For composite squarefree k, a(k) = (p^2 * m) such that (p^2 * m) is in A364996.
Permutation of the union of {1}, A001248, and A366825.

cp's OEIS Frontend

A366786 a(n) = A073481(n)*A005117(n).

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