A375731 -id:A375731 - cp's OEIS frontend

A375732 a(n) is the number of partitions of n having a cube number of parts whose sum of cubes is a cube.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 5, 2, 2, 1, 2, 2, 2, 4, 5, 9, 4, 5, 2, 6, 9, 9, 13, 12, 16, 8, 10, 8, 13, 19, 20, 26, 23, 23, 22, 22, 30, 38, 45, 47, 60, 54, 77, 87, 83, 89, 88, 104, 131, 156, 170, 202, 208, 220, 241

Offset: 0

Felix Huber, Aug 28 2024

nonn

			a(37) counts the 4 partitions [1, 1, 1, 2, 6, 8, 9, 9] with 8 = 2^3 parts and 1^3 + 1^3 + 1^3 + 2^3 + 6^3 + 8^3 + 9^3 + 9^3 = 13^3, [1, 1, 2, 4, 4, 6, 8, 11] with 8 = 2^3 parts and 1^3 + 1^3 + 2^3 + 4^3 + 4^3 + 6^3 + 8^3 + 11^3 = 13^3, [1, 1, 1, 2, 2, 2, 10, 18] with 8 = 2^3 parts and 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 10^3 + 18^3 = 19^3, [37] with 1 = 1^3 part and 37^3 = 37^3.

Cf. A003108, A025457, A240127, A240128, A259792, A375731.

# first Maple program to calculate the sequence:
A375732:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if type(root(numelems(i),3),integer) and type(root(add(i[j]^3,j=1..nops(i)),3), integer) then a:=a+1 fi od; return a end proc; seq(A375732(n),n=0..75);
# second Maple program to calculate the partitions:
A375732part:=proc(n) local L,i,j; L:=[]; for i in combinat:-partition(n) do if type(root(numelems(i),3),integer) and type(root(add(i[j]^3,j=1..nops(i)), 3),integer) then L:=[op(L),i] fi od; return op(L); end proc; A375732part(37);

a(n) = my(nb=0); forpart(p=n, if (ispower(#p,3) && ispower(sum(k=1, #p, p[k]^3),3), nb++)); nb; \\ Michel Marcus, Sep 01 2024

1 <= a(n) <= A240128(n).

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + 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

Felix Huber, Apr 04 2025

nonn

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.

			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.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the partitions

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

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);

cp's OEIS Frontend

A375732 a(n) is the number of partitions of n having a cube number of parts whose sum of cubes is a cube.

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

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cp's OEIS Frontend

A375732 a(n) is the number of partitions of n having a cube number of parts whose sum of cubes is a cube.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

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