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

1,8

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

Table of n, a(n) for n=1..100.

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

Crossrefs

Cf. A064047, A333524, A333525, A333526, A333527, A333528, A335152.

Sequence in context: A203638 A040002 A151798 * A035684 A049111 A096509

Adjacent sequences: A334894 A334895 A334896 * A334898 A334899 A334900

Keyword

nonn

Author

Ya-Ping Lu, May 15 2020

Extensions

Terms a(51) and beyond from Andrew Howroyd, May 15 2020

Status

approved