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

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

1,2

Comments

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.

Links

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

Formula

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A175121 A133905 A131826 * A051961 A339794 A251544

Adjacent sequences: A366783 A366784 A366785 * A366787 A366788 A366789

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 16 2023

Status

approved