cp's OEIS Frontend

A252849 Numbers with an even number of square divisors.

%I A252849 #58 Nov 18 2020 10:26:38
%S A252849 4,8,9,12,18,20,24,25,27,28,36,40,44,45,49,50,52,54,56,60,63,64,68,72,
%T A252849 75,76,84,88,90,92,98,99,100,104,108,116,117,120,121,124,125,126,128,
%U A252849 132,135,136,140,144,147,148
%N A252849 Numbers with an even number of square divisors.
%C A252849 Closed lockers in the locker problem where the student numbers are the set of perfect squares.
%C A252849 The locker problem is a classic mathematical problem. Imagine a row containing an infinite number of lockers numbered from one to infinity. Also imagine an infinite number of students numbered from one to infinity. All of the lockers begin closed. The first student opens every locker that is a multiple of one, which is every locker. The second student closes every locker that is a multiple of two, so all of the even-numbered lockers are closed. The third student opens or closes every locker that is a multiple of three. This process continues for all of the students.
%C A252849 A variant on the locker problem is when not all student numbers are considered; in the case of this sequence, only the square-numbered students open and close lockers. The sequence here is a list of the closed lockers after all of the students have gone.
%C A252849 From _Amiram Eldar_, Jul 07 2020: (Start)
%C A252849 Numbers k such that the largest square dividing k (A008833) is not a fourth power.
%C A252849 The asymptotic density of this sequence is 1 - Pi^2/15 = 1 - A182448 = 0.342026... (Cesàro, 1885). (End)
%C A252849 Closed under application of A331590: for n, k >= 1, A331590(a(n), k) is in the sequence. - _Peter Munn_, Sep 18 2020
%H A252849 Reinhard Zumkeller, <a href="/A252849/b252849.txt">Table of n, a(n) for n = 1..10000</a>
%H A252849 Ernest Cesàro, <a href="https://doi.org/10.1007/BF02420801">Le plus grand diviseur carré</a>, Annali di Matematica Pura ed Applicata, Vol. 13, No. 1 (1885), pp. 251-268, <a href="https://iris.univ-lille.fr/handle/1908/1932">entire volume</a>.
%H A252849 K. A. P. Dagal, <a href="http://arxiv.org/abs/1307.6455">Generalized Locker Problem</a>, arXiv:1307.6455 [math.NT], 2013.
%H A252849 B. Torrence and S. Wagon, <a href="https://cms.math.ca/crux/v33/n4/page232-236.pdf">The Locker Problem</a>, Crux Mathematicorum, 2007, 33(4), 232-236.
%F A252849 From _Peter Munn_, Sep 18 2020: (Start)
%F A252849 Numbers k such that A046951(k) mod 2 = 0.
%F A252849 Numbers k such that A335324(k) > 1.
%F A252849 (End)
%t A252849 Position[Length@ Select[Divisors@ #, IntegerQ@ Sqrt@ # &] & /@ Range@ 150, _Integer?EvenQ] // Flatten (* _Michael De Vlieger_, Mar 23 2015 *)
%o A252849 (C++)
%o A252849 #include <iostream>
%o A252849 using namespace std;
%o A252849 int main()
%o A252849 {
%o A252849   const int one_k = 1000;
%o A252849   //all numbers in sequence up to one_k are given
%o A252849   int lockers [one_k] = {};
%o A252849   int A = 0;
%o A252849   while (A < one_k) {
%o A252849     lockers [A] = A+1;
%o A252849      A = A + 1;
%o A252849   }
%o A252849   int B = 1;
%o A252849   while ( ((B+1) * (B+1)) <= one_k) {
%o A252849     int C = ((B+1) * (B+1));
%o A252849     int D = one_k/C;
%o A252849     int E = 1;
%o A252849     while (E <= D) {
%o A252849       lockers [(C*E)-1] = -1 * lockers [(C*E)-1];
%o A252849       E = E + 1;
%o A252849     }
%o A252849     B = B + 1;
%o A252849   }
%o A252849   int F = 0;
%o A252849   while (F < one_k) {
%o A252849     if (lockers [F] < 0) {
%o A252849       cout << (-1 * lockers [F]) << endl;
%o A252849     }
%o A252849     F = F + 1;
%o A252849   }
%o A252849   return 0;
%o A252849 }  /* _Walker Dewey Anderson_, Mar 22 2015 */
%o A252849 (PARI) isok(n) = sumdiv(n, d, issquare(d)) % 2 == 0; \\ _Michel Marcus_, Mar 22 2015
%o A252849 (Haskell)
%o A252849 a252849 n = a252849_list !! (n-1)
%o A252849 a252849_list = filter (even . a046951) [1..]
%o A252849 -- _Reinhard Zumkeller_, Apr 06 2015
%Y A252849 Complement of A252895.
%Y A252849 A046951, A335324 are used in a formula defining this sequence.
%Y A252849 Disjoint union of A336593 and A336594.
%Y A252849 A030140, A038109, A082293, A217319 are subsequences.
%Y A252849 Cf. A000290, A008833, A182448, A331590.
%Y A252849 Ordered 3rd trisection of A225546.
%K A252849 nonn
%O A252849 1,1
%A A252849 _Walker Dewey Anderson_, Mar 22 2015