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.
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, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 23, 24, 24, 25, 25, 25, 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
Offset: 1
Keywords
Examples
a(1) = 0 because 1 is the only positive integer <= 1. a(2) = 0 because 1 and 2 are the only two positive integers <= 2, and sqrt(1*2) is not an integer. a(4) = 1 because 2 = sqrt(1*4). a(8) = 2 because 2 = sqrt(1*4) and 4 = sqrt(2*8). a(9) = 4 because 2 = sqrt(1*4), 3 = sqrt(1*9), 4 = sqrt(2*8), and 6 = sqrt(4*9). 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).
Links
- Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020. See Table 3.1.
Programs
-
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 -
Python
list1 = [] list2 = [] nmax = 100 for i in range(1, nmax+1): list1.append(i*i) for j in range(1, i+1): for k in range(j+1, i+1): m = j*k if m in list1: list1.remove(m) list2.append(m) print(i, len(list2))
Formula
a(n) = n - A064047(n).
Extensions
Terms a(51) and beyond from Andrew Howroyd, May 15 2020
Comments