A210490 Union of positive squares (A000290 \ {0}) and squarefree numbers (A005117).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94

Offset: 1

Author

Reinhard Zumkeller, Jan 24 2013

Keywords

• nonn

Comments

Numbers n such that either all exponents in the prime factorization of n (cf. A124010) are even or all are = 1.

Every positive integer can be expressed as the product of two elements of this sequence. Every integer > 1 can be expressed as the product of two distinct members of the sequence. - Franklin T. Adams-Watters, Apr 08 2016

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Square Number
• Eric Weisstein's World of Mathematics, Squarefree
• Wikipedia, Squarefree integer
• Wikipedia, Square number

Crossrefs

Cf. A051144 (complement).

Cf. A000290, A005117.

Cf. A008966, A010052.

Programs

Haskell

a210490 n = a210490_list !! (n-1)
a210490_list = filter chi [1..] where
   chi x = all (== 1) es || all even es where es = a124010_row x

PARI

isok(m) = issquare(m) || issquarefree(m); \\ Michel Marcus, Feb 03 2022

Python

from math import isqrt
from sympy import mobius
def A210490(n):
    def f(x): return int(n+1+x-(y:=isqrt(x))-sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Mar 23 2025

Formula

A008966(a(n)) + A010052(a(n)) > 0.

Extensions

A more precise name from Michel Marcus, Feb 03 2022