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A334897 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.

%I A334897 #40 May 27 2020 02:42:55
%S A334897 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 A334897 15,15,15,15,16,16,16,16,16,16,16,16,17,17,17,17,18,23,24,24,25,25,25,
%U A334897 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 A334897 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 A334897 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 A334897 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 A334897 a(n) = n - A064047(n).
%e A334897 a(1) = 0 because 1 is the only positive integer <= 1.
%e A334897 a(2) = 0 because 1 and 2 are the only two positive integers <= 2, and sqrt(1*2) is not an integer.
%e A334897 a(4) = 1 because 2 = sqrt(1*4).
%e A334897 a(8) = 2 because 2 = sqrt(1*4) and 4 = sqrt(2*8).
%e A334897 a(9) = 4 because 2 = sqrt(1*4), 3 = sqrt(1*9), 4 = sqrt(2*8), and 6 = sqrt(4*9).
%e A334897 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 A334897 (Python)
%o A334897 list1 = []
%o A334897 list2 = []
%o A334897 nmax = 100
%o A334897 for i in range(1, nmax+1):
%o A334897     list1.append(i*i)
%o A334897     for j in range(1, i+1):
%o A334897         for k in range(j+1, i+1):
%o A334897             m = j*k
%o A334897             if m in list1:
%o A334897                 list1.remove(m)
%o A334897                 list2.append(m)
%o A334897     print(i, len(list2))
%o A334897 (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 A334897 Cf. A064047, A333524, A333525, A333526, A333527, A333528, A335152.
%K A334897 nonn
%O A334897 1,8
%A A334897 _Ya-Ping Lu_, May 15 2020
%E A334897 Terms a(51) and beyond from _Andrew Howroyd_, May 15 2020