A335152 -id:A335152 - cp's OEIS frontend

A064047 Number of numbers only appearing once in 1-to-n multiplication table.

1, 2, 3, 3, 4, 5, 6, 6, 5, 6, 7, 8, 9, 10, 11, 10, 11, 12, 13, 13, 14, 15, 16, 17, 15, 16, 15, 15, 16, 17, 18, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 30, 26, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43

Offset: 1

Matthew Somerville (matthew.somerville(AT)trinity.oxford.ac.uk), Aug 24 2001

nonn

For n <= 127, this is the same as the number of vertices of the polytope representing the number n. The latter is given in A335152. The sequences differ starting at n = 128. See A335152 and Lu and Deng, Appendix. - N. J. A. Sloane, May 25 2020

a(n) is the number of x in [1,n] such that x^2 has no divisor d with x < d <= n. - Robert Israel, Sep 03 2020

			In the 1-to-5 multiplication table, four numbers (1,9,16,25) appear once only. Therefore a(5)=4.

Robert Israel, Table of n, a(n) for n = 1..10000
Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020.

Cf. A064048, A057142, A057143, A057144, A335152.

N:= 200: # for a(1)..a(N)
V:= Vector(N):
for x from 1 to N do
  y:= min(N, min(select(`>`,numtheory:-divisors(x^2),x))-1);
  V[x..y]:= map(`+`,V[x..y],1)
od:
convert(V,list); # Robert Israel, Sep 03 2020

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

N. J. A. Sloane, Mar 26 2020

Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020. See Table 3.1.

Cf. A333524-A333528, A335152.

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

Ya-Ping Lu, May 15 2020

nonn

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.

			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).

Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020. See Table 3.1.

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

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

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))

a(n) = n - A064047(n).

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

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A064047 Number of numbers only appearing once in 1-to-n multiplication table.

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A333524 Normalized volume of polytope representing the number n.

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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.

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