A338687 -id:A338687 - cp's OEIS frontend

A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

1, 2, 2, 3, 3, 3, 3, 2, 3, 4, 4, 4, 3, 2, 3, 4, 3, 5, 5, 2, 4, 3, 2, 3, 4, 5, 4, 4, 4, 4, 2, 2, 6, 5, 2, 6, 8, 6, 4, 5, 6, 6, 5, 4, 6, 5, 4, 5, 4, 10, 6, 5, 8, 3, 5, 5, 7, 6, 4, 5, 7, 5, 2, 6, 7, 6, 7, 8, 6, 4, 5, 6, 8, 6, 2, 4, 8, 4, 6, 3, 5, 10, 6, 8, 7, 5, 5, 6, 5, 5, 5, 7, 6, 4, 4, 6, 3, 8, 7, 7

Offset: 1

Zhi-Wei Sun, Apr 23 2021

nonn

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..7*10^6.
See also A338687 for a similar conjecture.
Conjecture verified up to 2*10^9. - Giovanni Resta, Apr 28 2021

			a(1) = 1 with 1 = 0^5 + 1^2 + [2^2/7].
a(166) = 1 with 166 = 0^5 + 1^2 + [34^2/7].
a(323) = 1 with 323 = 2^5 + 17^2 + [4^2/7].
a(815) = 1 with 815 = 2^5 + 1^2 + [74^2/7].
a(2069) = 1 with 2069 = 0^5 + 37^2 + [70^2/7].
a(7560) = 1 with 7560 = 2^5 + 64^2 + [155^2/7].
a(24195) = 1 with 24195 = 0^5 + 8^2 + [411^2/7].
a(90886) = 2 with 90886 = 4^5 + 34^2 + [788^2/7] = 9^5 + 139^2 + [296^2/7].

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000584, A270920, A338687, A343387, A343391, A343397.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-x^5-Floor[y^2/7]],r=r+1],{x,0,(n-1)^(1/5)},{y,2,Sqrt[7(n-x^5)-1]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A338696 Number of ways to write n as x^3 + y^2 + z(3z+2), where x and y are nonnegative integers, and z is an integer.

Original entry on oeis.org

3, 3, 1, 1, 3, 3, 1, 2, 6, 5, 1, 2, 3, 2, 1, 3, 8, 4, 0, 2, 3, 4, 1, 3, 7, 4, 2, 3, 3, 3, 3, 4, 7, 4, 2, 4, 5, 5, 1, 2, 7, 5, 3, 6, 5, 1, 2, 3, 7, 5, 2, 6, 2, 2, 1, 2, 10, 5, 2, 4, 2, 1, 1, 7, 11, 8, 2, 5, 6, 5, 3, 4, 11, 3, 1, 5, 5, 2, 1, 5, 8, 6, 4, 5, 5, 5, 3, 2, 9, 7, 2, 6, 4, 5, 1, 5, 10, 5, 2, 4

Offset: 1

Zhi-Wei Sun, Apr 24 2021

nonn

Conjecture: a(n) > 0 except for n = 19.
We have verified this for n up to 5*10^6.
As z*(3*z+2) = floor((3*z+1)^2/3) and 19 = 0^3 + 4^2 + floor(3^2/3), the conjecture implies that each n = 0,1,... can be written as x^3 + y^2 + floor(z^2/3) with x,y,z nonnegative integers.

			a(63) = 1 with 63 = 3^3 + 6^2 + 0*(3*0+2).
a(327) = 1 with 327 = 5^3 + 13^2 + 3*(3*3+2).
a(478) = 1 with 478 = 6^3 + 1^2 + 9*(3*9+2).
a(847) = 1 with 847 = 1^3 + 29^2 + 1*(3*1+2).
a(1043) = 1 with 1043 = 3^3 + 20^2 + 14*(3*14+2).
a(3175) = 1 with 3175 = 5^3 + 35^2 + (-25)*(3*(-25)+2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A001082, A262813, A270469, A338686, A338687.

OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]];
tab={};Do[r=0;Do[If[OctQ[n-x^3-y^2],r=r+1],{x,0,n^(1/3)},{y,0,Sqrt[n-x^3]}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

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A338696 Number of ways to write n as x^3 + y^2 + z(3z+2), where x and y are nonnegative integers, and z is an integer.

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cp's OEIS Frontend

A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

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Mathematica

A338696 Number of ways to write n as x^3 + y^2 + z*(3*z+2), where x and y are nonnegative integers, and z is an integer.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

A338696 Number of ways to write n as x^3 + y^2 + z(3z+2), where x and y are nonnegative integers, and z is an integer.