cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A333524 Normalized volume of polytope representing the number n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 6, 8, 8, 9, 9, 11, 14, 16, 16, 18, 18, 20, 24, 28, 28, 29, 38, 44, 52, 58, 58, 59, 59, 63, 73, 83, 100, 102, 102, 115, 131, 135, 135, 139, 139, 148, 159, 176, 176, 178, 216, 226
Offset: 1

Views

Author

N. J. A. Sloane, Mar 26 2020

Keywords

Crossrefs

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.

Original entry on oeis.org

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

Views

Author

Ya-Ping Lu, May 15 2020

Keywords

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.

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).
		

Crossrefs

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
Showing 1-2 of 2 results.