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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.
1
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
1,8
COMMENTS
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.
LINKS
Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020. See Table 3.1.
FORMULA
a(n) = n - A064047(n).
EXAMPLE
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).
PROG
(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))
(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
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, May 15 2020
EXTENSIONS
Terms a(51) and beyond from Andrew Howroyd, May 15 2020
STATUS
approved