A306383 -id:A306383 - cp's OEIS frontend

A306382 Positive integers not representable as Pen(x) + Pen(y) + 2Pen(z), where x, y, z are nonnegative integers, and Pen(k) denotes the pentagonal number k(3k-1)/2.

9, 18, 21, 28, 31, 39, 43, 55, 69, 74, 89, 90, 98, 109, 111, 113, 134, 135, 144, 193, 202, 214, 230, 243, 260, 265, 273, 275, 310, 510, 553, 698, 699, 749, 773, 780, 865, 878, 945, 965, 1219, 1398, 1413, 2153, 2168, 2335, 2828, 3178, 3793

Offset: 1

Zhi-Wei Sun, Feb 11 2019

nonn

Conjecture 1: This sequence only has 49 terms as listed.
Conjecture 2: Any integer n > 6471 can be written as Pen(x) + Pen(y) + 3*Pen(z) with x,y,z nonnegative integers. Any integer n > 7727 can be written as Pen(x) + 2*Pen(y) + 2*Pen(z) with x,y,z nonnegative integers. Also, any integer n > 4451 can be written as Pen(x) + 2*Pen(y) + 4*Pen(z) with x,y,z nonnegative integers.
Conjecture 3: Let N(1) = 5928, N(4) = 2761, N(8) = 8868 and N(11) = 9929. For each c among 1, 4, 8, 11, any integer n > N(c) can be written as x*(3x+1)/2 + y*(3y+1) + c*z*(3z+1)/2 with x,y,z nonnegative integers.
Conjecture 4: Any integer n > 5544 can be written as x*(3x+1)/2 + y*(3y+1)/2 + 3z*(3z+1)/2 with x,y,z nonnegative integers. Any integer n > 7093 can be written as x*(3x+1)/2 + 3*y*(3y+1)/2 + 2z*(3z+1) with x,y,z nonnegative integers. Also, any integer n > 8181 can be written as x*(3x+1)/2 + 2y*(3y+1) + 3z*(3z+1) with x,y,z nonnegative integers.
Conjecture 5: For each positive integer m, there are only finitely many positive integers not representable as x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 with x,y,z in the set {m, m+1, ...}.
See also A306383 for similar conjectures.

			a(1) = 9 since the set {Pen(x) + Pen(y) + 2*Pen(z): x,y,z = 0,1,2,...} contains 1..8 but it does not contain 9.

Georg Fischer, Table of n, a(n) for n = 1..49
Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.

Cf. A000217, A000326, A005449, A008443, A306383.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);
tab={};Do[Do[If[PenQ[m-x(3x-1)-y(3y-1)/2],Goto[aa]],{x,0,(Sqrt[12m+1]+1)/6},{y,0,(Sqrt[12(m-x(3x-1))+1]+1)/6}];tab=Append[tab,m];Label[aa],{m,1,5000}];Print[tab]

A306439 Number of ways to write n as x(3x+1)/2 + y(3y+1)/2 + z(3z+1) + 3w(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 15 2019

nonn

Conjecture 1: a(n) > 0 for all n > 5, and a(n) = 1 only for n = 0, 2, 7, 9, 11, 12, 16, 31, 33, 41.
Conjecture 2: Let n be any integer greater than 9, and let p(x) denote x*(3x+1)/2. For each c = 2, 4, 9, we can write n as p(x) + 2*p(y) + 3*p(z) + c*p(w) with x,y,z,w nonnegative integers.
See also Conjecture 5.2 of the linked 2016 paper.

			a(12) = 1 with 12 = 0*(3*0+1)/2 + 1*(3*1+1)/2 + 1*(3*1+1) + 3*1*(3*1+1)/2.
a(31) = 1 with 31 = 1*(3*1+1)/2 + 3*(3*3+1)/2 + 2*(3*2+1) + 3*0*(3*0+1)/2.
a(33) = 1 with 33 = 2*(3*2+1)/2 + 4*(3*4+1)/2 + 0*(3*0+1) + 3*0*(3*0+1)/2.
a(41) = 1 with 41 = 3*(3*3+1)/2 + 4*(3*4+1)/2 + 0*(3*0+1) + 3*0*(3*0+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.

Cf. A005449, A306382, A306383.

PQ[n_]:=PQ[n]=IntegerQ[Sqrt[24n+1]]&&Mod[Sqrt[24n+1],6]==1;
tab={};Do[r=0;Do[If[PQ[n-3x(3x+1)/2-y(3y+1)-z(3z+1)/2],r=r+1],{x,0,(Sqrt[8n+1]-1)/6},{y,0,(Sqrt[12(n-3x(3x+1)/2)+1]-1)/6},{z,0,(Sqrt[12(n-3x(3x+1)/2-y(3y+1))+1]-1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]

A377224 Number of ways to write n as x(5x+1) + y(5y+1)/2 + z(5z+1)/2, where x,y,z are integers with y(5y+1) <= z(5z+1).

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Nov 13 2024

nonn

Conjecture 1: a(n) = 0 only for n = 1. Also, a(n) = 1 only for n = 0, 2, 3, 5, 7, 14, 16, 19, 37, 43, 58, 61, 79.
This has been verified for n <= 2*10^6.
Conjecture 2: Let N be the set of all nonnegative integers. Then
{x*(5*x+1) + y*(5*y+1)/2 + 5*z*(5*z+1)/2: x,y,z are integers} = N\{1,5},
{x*(5*x+1) + y*(5*y+1)/2 + 3*z*(5*z+1)/2: x,y,z are integers} = N\{1,5,32},
{x*(5*x+1) + y*(5*y+1)/2 + 2*z*(5*z+1): x,y,z are integers} = N\{1,5,70},
and
{x*(5*x+1)/2 + y*(5*y+1)/2 + z*(5*z+1)/2: x,y,z are integers} = N\{1,10,19,94}.
Conjecture 3: We have
{x*(5*x+3) + y*(5*y+3)/2 + 3*z*(5*z+3)/2: x,y,z are integers} = N\{31,77},
{x*(5*x+3) + y*(5*y+3)/2 + 5*z*(5*z+3): x,y,z are integers} = N\{10,16},
and
{x*(5*x+3)/2 + y*(5*y+3)/2 + 5*z*(5*z+3)/2: x,y,z are integers} = N\{3,15,29,44}.

			a(14) = 1 with 14 = 0*(5*0+1) + 1*(5*1+1)/2 + 2*(5*2+1)/2.
a(37) = 1 with 37 = (-1)*(5*(-1)+1) + (-2)*(5*(-2)+1)/2 + 3*(5*3+1)/2.
a(58) = 1 with 58 = (-2)*(5*(-2)+1) + (-1)*(5*(-1)+1)/2 + (-4)*(5*(-4)+1)/2.
a(79) = 1 with 79 = -4*(5*(-4)+1) + 0*(5*0+1)/2 + 1*(5*1+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162 (2016), 190-211.
Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math. 63 (2020), 501-520.
Zhi-Wei Sun, New results similar to Lagrange's four-square theorem, arXiv:2411.14308 [math.NT], 2024.

Cf. A057569, A085787, A306383.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[40(n-x(5x+1)-y(5y+1)/2)+1],r=r+1],{x,-Floor[(Sqrt[20n+1]+1)/10],(Sqrt[20n+1]-1)/10},{y,-Floor[(Sqrt[20(n-x(5x+1))+1]+1)/10],Floor[(Sqrt[20(n-x(5x+1))+1]-1)/10]}];tab=Append[tab,r],{n,0,100}];Print[tab]

cp's OEIS Frontend

A306382 Positive integers not representable as Pen(x) + Pen(y) + 2Pen(z), where x, y, z are nonnegative integers, and Pen(k) denotes the pentagonal number k(3k-1)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A306439 Number of ways to write n as x(3x+1)/2 + y(3y+1)/2 + z(3z+1) + 3w(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A377224 Number of ways to write n as x(5x+1) + y(5y+1)/2 + z(5z+1)/2, where x,y,z are integers with y(5y+1) <= z(5z+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A306382 Positive integers not representable as Pen(x) + Pen(y) + 2*Pen(z), where x, y, z are nonnegative integers, and Pen(k) denotes the pentagonal number k*(3k-1)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A306439 Number of ways to write n as x*(3x+1)/2 + y*(3y+1)/2 + z*(3z+1) + 3w*(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A377224 Number of ways to write n as x*(5*x+1) + y*(5*y+1)/2 + z*(5*z+1)/2, where x,y,z are integers with y*(5*y+1) <= z*(5*z+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A306382 Positive integers not representable as Pen(x) + Pen(y) + 2Pen(z), where x, y, z are nonnegative integers, and Pen(k) denotes the pentagonal number k(3k-1)/2.

A306439 Number of ways to write n as x(3x+1)/2 + y(3y+1)/2 + z(3z+1) + 3w(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

A377224 Number of ways to write n as x(5x+1) + y(5y+1)/2 + z(5z+1)/2, where x,y,z are integers with y(5y+1) <= z(5z+1).