A064047 -id:A064047 - cp's OEIS frontend

A335152 Number of vertices in polytope representing the number n.

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, 44, 44, 44, 45, 46, 47, 48, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 58, 57, 55, 56, 57, 58, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 69, 68, 69, 70, 71, 66, 67, 68, 68, 66, 67, 68, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

N. J. A. Sloane, May 25 2020, based on correspondence with Ya-Ping Lu

nonn

More than the usual number of terms are shown here in order to distinguish this sequence from A064047. The two sequences first differ at n=128.
Comments from Ya-Ping Lu, May 25 2020 (Start):
Concerning the sequences A064047 and the number of vertices of the polytope representing n (the present sequence). These two sequences are similar but not exactly the same.
As you can see from the pdf file attached, for n<=127, A064047 is the same as N_vert. For n > =128, A064047 is always greater than or equal to N_vert. This is due to the fact that in some cases not all the non-vertex numbers on the polytope can be written as the geometric mean of two integers on the polytope. See also A334810 and A334897.
The cases where A064047 is greater than N_vert are highlighted in yellow in the attached file. (End)

Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020.
Ya-Ping Lu, Table comparing A064047 (column 2) and a(n) (column 3). [Yellow lines show where the values are different.]

Cf. A064047, A333524, A334810, A334897.

A064048 Number of most frequently occurring numbers in the 1-to-n multiplication table.

Original entry on oeis.org

1, 1, 3, 1, 1, 2, 2, 4, 5, 9, 9, 2, 2, 2, 4, 5, 5, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 7, 7, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 1, 1, 1, 1, 1, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

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

nonn

			In the 1-to-6 multiplication table, the most frequently occurring numbers (each occurring 4 times) are 6 and 12. Therefore a(6)=2.

Branden Aldridge, Table of n, a(n) for n = 1..20000

Cf. A064047, A057142, A057143, A057144.

mfon[n_]:=Module[{x=SortBy[Tally[Times@@@Tuples[Range[n],2]], Last]}, Length[ Select[x,#[[2]]==x[[-1,2]]&]]]; Array[mfon,100] (* Harvey P. Dale, Oct 20 2012 *)

a(n)=my(v=List(),ct,s,r,t); for(a=1,n,for(b=1,n,listput(v,a*b))); s=Set(v); for(i=1,#s, t=sum(j=1,#v,v[j]==s[i]); if(tr,ct=1;r=t, ct++)); ct \\ Charles R Greathouse IV, Feb 05 2022

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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A335152 Number of vertices in polytope representing the number n.

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A064048 Number of most frequently occurring numbers in the 1-to-n multiplication table.

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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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