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

%I A210490 #25 Mar 23 2025 13:40:43
%S A210490 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,
%T A210490 34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,65,66,
%U A210490 67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,91,93,94
%N A210490 Union of positive squares (A000290 \ {0}) and squarefree numbers (A005117).
%C A210490 Numbers n such that either all exponents in the prime factorization of n (cf. A124010) are even or all are = 1.
%C A210490 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
%H A210490 Reinhard Zumkeller, <a href="/A210490/b210490.txt">Table of n, a(n) for n = 1..10000</a>
%H A210490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A210490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A210490 Wikipedia, <a href="http://en.wikipedia.org/wiki/Squarefree_integer">Squarefree integer</a>
%H A210490 Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_number">Square number</a>
%F A210490 A008966(a(n)) + A010052(a(n)) > 0.
%o A210490 (Haskell)
%o A210490 a210490 n = a210490_list !! (n-1)
%o A210490 a210490_list = filter chi [1..] where
%o A210490    chi x = all (== 1) es || all even es where es = a124010_row x
%o A210490 (PARI) isok(m) = issquare(m) || issquarefree(m); \\ _Michel Marcus_, Feb 03 2022
%o A210490 (Python)
%o A210490 from math import isqrt
%o A210490 from sympy import mobius
%o A210490 def A210490(n):
%o A210490     def f(x): return int(n+1+x-(y:=isqrt(x))-sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
%o A210490     m, k = n, f(n)
%o A210490     while m != k: m, k = k, f(k)
%o A210490     return m # _Chai Wah Wu_, Mar 23 2025
%Y A210490 Cf. A051144 (complement).
%Y A210490 Cf. A000290, A005117.
%Y A210490 Cf. A008966, A010052.
%K A210490 nonn
%O A210490 1,2
%A A210490 _Reinhard Zumkeller_, Jan 24 2013
%E A210490 A more precise name from _Michel Marcus_, Feb 03 2022