|
%I #40 May 27 2020 02:42:55
%S 0,0,0,1,1,1,1,2,4,4,4,4,4,4,4,6,6,6,6,7,7,7,7,7,10,10,12,13,13,13,13,
%T 15,15,15,15,16,16,16,16,16,16,16,16,17,17,17,17,18,23,24,24,25,25,25,
%U 25,25,25,25,25,25,25,25,25,28,28,28,28,29,29,29,29,30,30,30,31,32,32,32,32,32,36,36,36,36,36,36,36,37,37,37,37,38,38,38,38,38,38,40,42,45
%N a(n) is the number of positive integers less than or equal to n that can be written as the geometric mean of two different positive integers less than or equal to n.
%C For n <= 127, a(n) = n - A335152. For n > 127, a(n) <= n - A335152. This sequence is nondecreasing, i.e., a(n) >= a(n-1) as the a(n) positive integers include all the a(n-1) positive integers for n-1.
%H Ya-Ping Lu and Shu-Fang Deng, <a href="http://arxiv.org/abs/2003.08968">Properties of Polytopes Representing Natural Numbers</a>, arXiv:2003.08968 [math.GM], 2020. See Table 3.1.
%F a(n) = n - A064047(n).
%e a(1) = 0 because 1 is the only positive integer <= 1.
%e a(2) = 0 because 1 and 2 are the only two positive integers <= 2, and sqrt(1*2) is not an integer.
%e a(4) = 1 because 2 = sqrt(1*4).
%e a(8) = 2 because 2 = sqrt(1*4) and 4 = sqrt(2*8).
%e a(9) = 4 because 2 = sqrt(1*4), 3 = sqrt(1*9), 4 = sqrt(2*8), and 6 = sqrt(4*9).
%e a(16) = 6 because 2 = sqrt(1*4), 3 = sqrt(1*9), 4 = sqrt(2*8), 6 = sqrt(4*9), 8 = sqrt(4*16), and 12 = sqrt(9*16).
%o (Python)
%o list1 = []
%o list2 = []
%o nmax = 100
%o for i in range(1, nmax+1):
%o list1.append(i*i)
%o for j in range(1, i+1):
%o for k in range(j+1, i+1):
%o m = j*k
%o if m in list1:
%o list1.remove(m)
%o list2.append(m)
%o print(i, len(list2))
%o (PARI) a(n)={sum(i=1, n, sum(j=1, i-1, i^2%j==0 && i^2/j<=n)>0)} \\ _Andrew Howroyd_, May 15 2020
%Y Cf. A064047, A333524, A333525, A333526, A333527, A333528, A335152.
%K nonn
%O 1,8
%A _Ya-Ping Lu_, May 15 2020
%E Terms a(51) and beyond from _Andrew Howroyd_, May 15 2020
|
