A295819 -id:A295819 - cp's OEIS frontend

A295820 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 <= n.

0, 2, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 13, 15, 15, 15, 17, 17, 17, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 31, 31, 31, 31, 35, 35, 35, 35, 35, 35

Offset: 0

Seiichi Manyama, Nov 28 2017

nonn

			Solutions to (x,y) = 1 and x^2 + y^2 <= 17;
  *         (1,4)
  * *       (1,3), (2,3)
  *   *     (1,2), (3,2)
* * * * *   (0,1), (1,1), (2,1), (3,1), (4,1)
  *         (1,0)
            a(17) = 11.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A049643, A224212, A295819, A295849.

a[n_] := Sum[Boole[GCD[i, j]==1 ], {i, 0, Sqrt[n]}, {j, 0, Sqrt[n-i^2]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

a(n) = {sum(i=0, sqrtint(n), sum(j=0, sqrtint(n-i^2), gcd(i, j) == 1))} \\ Andrew Howroyd, Dec 12 2017

a(n) = a(n-1) + A295819(n) for n > 0.

A295848 Number of nonnegative solutions to (x,y,z) = 1 and x^2 + y^2 + z^2 = n.

Original entry on oeis.org

0, 3, 3, 1, 0, 6, 3, 0, 0, 3, 6, 3, 0, 6, 6, 0, 0, 9, 3, 3, 0, 6, 3, 0, 0, 6, 12, 3, 0, 12, 6, 0, 0, 6, 9, 6, 0, 6, 9, 0, 0, 15, 6, 3, 0, 6, 6, 0, 0, 6, 12, 6, 0, 12, 9, 0, 0, 6, 6, 9, 0, 12, 12, 0, 0, 18, 12, 3, 0, 12, 6, 0, 0, 9, 18, 6, 0, 12, 6, 0, 0, 9, 9, 9

Offset: 0

Seiichi Manyama, Nov 29 2017

a(n)=0 for n in A047536. - Robert Israel, Nov 30 2017

			a(1) = 3;
(1,0,0) = 1 and 1^2 + 0^2 + 0^2 = 1.
(0,1,0) = 1 and 0^2 + 1^2 + 0^2 = 1.
(0,0,1) = 1 and 0^2 + 0^2 + 1^2 = 1.
a(2) = 3;
(1,1,0) = 1 and 1^2 + 1^2 + 0^2 = 2.
(1,0,1) = 1 and 1^2 + 0^2 + 1^2 = 2.
(0,1,1) = 1 and 0^2 + 1^2 + 1^2 = 2.
a(3) = 1;
(1,1,1) = 1 and 1^2 + 1^2 + 1^2 = 3.
a(5) = 6;
(2,1,0) = 1 and 2^2 + 1^2 + 0^2 = 5.
(2,0,1) = 1 and 2^2 + 0^2 + 1^2 = 5.
(1,2,0) = 1 and 1^2 + 2^2 + 0^2 = 5.
(1,0,2) = 1 and 1^2 + 0^2 + 2^2 = 5.
(0,2,1) = 1 and 0^2 + 2^2 + 1^2 = 5.
(0,1,2) = 1 and 0^2 + 1^2 + 2^2 = 5.

Robert Israel, Table of n, a(n) for n = 0..10000 (n=0..200 from Seiichi Manyama)

Cf. A002102, A047536, A048240, A295819, A295849.

N:= 100:
V:= Array(0..N):
for x from 0 to floor(sqrt(N/3)) do
  for y from x to floor(sqrt((N-x^2)/2)) do
    for z from y to floor(sqrt(N-x^2-y^2)) do
      if igcd(x,y,z) = 1 then
        r:= x^2 + y^2 + z^2;
        m:= nops({x,y,z});
        if m=3 then V[r]:= V[r]+6
        elif m=2 then V[r]:= V[r]+3
        else V[r]:= V[r]+1
        fi
      fi
od od od:
convert(V,list); # Robert Israel, Nov 30 2017

f[n_] := Total[ Length@ Permutations@# & /@ Select[ PowersRepresentations[n, 3, 2], GCD[#[[1]], #[[2]], #[[3]]] == 1 &]]; Array[f, 90, 0] (* Robert G. Wilson v, Nov 30 2017 *)

A295976 Number of nonnegative solutions to (x,y) = 1 and x^3 + y^3 = n.

Original entry on oeis.org

0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Seiichi Manyama, Dec 01 2017

nonn

Number of ordered pairs of two nonnegative natural numbers that are coprime and whose cubes add to n. - Antti Karttunen, May 31 2021

			For 1729, a(1729) = 4, because the following four ordered pairs, (1,12),  (9,10),  (10,9) and (12,1) satisfy the condition, as 1^3 + 12^3 = 9^3 + 10^3 = 1729. - _Antti Karttunen_, May 31 2021

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A173677, A225530, A295819, A295977.

Cf. also A202679.

{a(n) = sum(i=0, n, sum(j=0, n, if((gcd(i, j)==1) && (i^3+j^3==n), 1, 0)))}

A295976(n) = { my(s=0); for(i=0, oo, i3 = i^3; forstep(j=n-i3, 0, -1, if((i3+j^3==n) && gcd(i, j)==1, s++)); if(i3>n, return(s))); }; \\ Antti Karttunen, May 31 2021

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A295820 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 <= n.

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A295848 Number of nonnegative solutions to (x,y,z) = 1 and x^2 + y^2 + z^2 = n.

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A295976 Number of nonnegative solutions to (x,y) = 1 and x^3 + y^3 = n.

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