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 1-4 of 4 results.

A375580 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect cube.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 3, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 2, 2, 2, 2, 1, 2, 3, 2, 2, 3, 2, 0, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 1, 4, 2, 2, 2, 3, 1, 2, 3, 1, 3, 4, 1, 3, 2, 2, 1, 2, 2, 3, 3, 2, 4
Offset: 0

Views

Author

Felix Huber, Aug 19 2024

Keywords

Comments

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a cube.
Conjecture: for n > 176, a(n) > 0. - Charles R Greathouse IV, Aug 20 2024

Examples

			a(21) = 3 because the three partitions [1, 4, 16], [3, 6, 12], [7, 7, 7] satisfy the conditions: 1 + 4 + 16 = 21 and 1*4*16 = 4^3, 3 + 6 + 12 = 21 and 3*6*12 = 6^3, 7 + 7 + 7 = 21 and 7*7*7 = 7^3.
See also linked Maple code.

Links

Crossrefs

Cf. A338939, A375512, A375576, A103277, A069905.

Programs

  • Maple
    # See Huber link.
  • PARI
    a(n)=sum(x=1,n\3, sum(y=x,(n-x)\2, ispower(x*y*(n-x-y),3))) \\ Charles R Greathouse IV, Aug 20 2024
  • PARI
    \\ See Corneth link
  • Python
    from sympy import integer_nthroot
def A375580(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if integer_nthroot((n-x-y-2)*(x+1)*(y+1),3)[1]) # Chai Wah Wu, Aug 21 2024

Formula

Trivial upper bound: a(n) <= A069905(n). - Charles R Greathouse IV, Aug 23 2024

A375576 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 4, 1, 2, 3, 2, 2, 4, 2, 4, 3, 4, 2, 5, 4, 2, 6, 5, 2, 8, 4, 8, 4, 4, 5, 10, 5, 3, 8, 7, 6, 12, 5, 6, 7, 6, 7, 11, 5, 6, 8, 12, 6, 11, 8, 11, 11, 6, 3, 22, 6, 12, 12, 8, 9, 13, 12, 7, 14, 14, 6, 18, 7, 7, 18, 13, 14, 13, 7, 19, 10
Offset: 0

Views

Author

Felix Huber, Aug 19 2024

Keywords

Comments

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a square cuboid with height 1.

Examples

			a(24) = 4 because the four partitions [2, 4, 18], [3, 9, 12], [4, 4, 16], [4, 10, 10] satisfy the conditions: 2 + 4 + 18 = 24 and 2*4*18 = 12^2, 3 + 9 + 12 = 24 and 3*9*12 = 18^2, 4 + 4 + 16 = 24 and 4*4*16 = 16^2, 4 + 10 + 10 = 24 and 4*10*10 = 20^2.
See also linked Maple code.

Links

Crossrefs

Cf. A338939, A375512, A375580.

Programs

  • Maple
    See Huber link.
  • Python
    from sympy.ntheory.primetest import is_square
def A375567(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if is_square((n-x-y-2)*(x+1)*(y+1))) # Chai Wah Wu, Aug 22 2024

A338940 a(n) is the number of solutions to the Diophantine equation p * x * (x + n) = y^2 with p = a*b a perfect square and a+b = n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 1, 4, 2, 1, 2, 0, 3, 0, 1, 0, 4, 4, 3, 0, 1, 1, 12, 0, 3, 0, 3, 4, 2, 1, 1, 4, 12, 1, 4, 0, 1, 7, 1, 0, 7, 0, 10, 4, 3, 1, 3, 4, 4, 0, 3, 0, 12, 1, 1, 0, 4, 16, 4, 0, 3, 0, 12, 0, 7, 1, 3, 14, 1, 0, 12, 0, 21, 0, 3, 0, 4, 16, 1, 4, 4, 1, 21, 4, 1, 0, 1, 4, 10, 1, 2, 0, 10
Offset: 1

Views

Author

Hein van Winkel, Nov 16 2020

Keywords

Comments

Related to Heron triangles with a partition point on a side of length n where the incircle is tangent. Some partitions correspond to a finite number of Heron triangles. The numbers a(n) in this sequence are the numbers of Heron triangles that match these 'finite' partitions.

Examples

			n = 25 = 5 + 20 = 9 + 16 gives 100 * x * (x + 25) = y^2 or 144 * x * (x + 25) = y^2 or 144 * x * (x + 25) = y^2. And the solutions are (x,y) = (144,1560) or (20,300) or (144,1872) or (20,360).

Links

Crossrefs

Cf. A115878, A338939.

Formula

Let n = 2^t * p_1^a_1 * p_2^a_2 *...* p_r^a_r * q_1^b_1 * q_2^b_2 *...* q_s^b_s with t>=0, a_i>=0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j = -1 (mod 4) for j=1..s.
Further let A = (2a_1 + 1) * (2a_2 + 1) *...* (2a_r + 1) and B = A * (2b_1 + 1) * (2b_2 + 1) *...* (2b_s + 1).
Then a(n) = (A-1) * (B-1) / 4 for t = 0 and a(n) = A * (B-1) / 2 for t = 1 AND t = 2 and a(n) = (2*t - 3) * A * (B+1) / 2 for t > 2.
a(n) = A338939(n) * A115878(n).

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 0, 5, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 1, 4, 2, 7, 4, 4, 0, 3, 5, 3, 4, 2, 1, 7, 2, 1, 5, 9, 3, 5, 3, 4, 1, 9, 2, 6, 3, 5, 6, 5, 4, 7, 5, 1, 5, 6, 3, 13, 7, 8, 4, 6, 0, 4, 4, 11, 5, 13, 2
Offset: 1

Views

Author

Felix Huber, Apr 04 2025

Keywords

Comments

a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.
Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.

Examples

			The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.

Links

Crossrefs

Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.

Programs

  • Maple
    A382407:=proc(n)
    local a,x,y,z;
    a:=0;
    for x to n/3 do
        for y from x to (n-x)/2 do
            z:=n-x-y;
            if issqr(x*y+x*z+y*z) then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A382407(n),n=1..87);
Showing 1-4 of 4 results.

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