A343397 -id:A343397 - cp's OEIS frontend

A343387 Number of ways to write n as x^2 + [y^2/2] + [z^4/8], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

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

Offset: 0

Zhi-Wei Sun, Apr 13 2021

nonn

Conjecture: a(n) > 0 for all n >= 0.
This has been verified for all n = 0..10^5.
We also conjecture that each n = 0,1,... can be written as x^2 + [y^2/3] + [z^4/7] with x,y,z nonnegative integers.
See also A343391 for a similar conjecture.

			a(0) = 1 with 0 = 0^2 + [1^2/2] + [1^4/8].
a(47) = 1 with 47 = 5^2 + [5^2/2] + [3^4/8].
a(55) = 1 with 55 = 7^2 + [3^2/2] + [2^4/8].
a(217) = 1 with 217 = 11^2 + [6^2/2] + [5^4/8].
a(377) = 1 with 377 9^2 + [23^2/2] + [4^4/8].
a(392) = 1 with 392 = 0^2 + [28^2/2] + [1^4/8].
a(734) = 1 with 734 = 12^2 + [32^2/2] + [5^4/8].
a(1052) = 1 with 1052 = 32^2 + [6^2/2] + [3^4/8].
a(1054) = 1 with 1054 = 30^2 + [17^2/2] + [3^4/8].
a(1817) = 1 with 1817 = 39^2 + [23^2/2] + [4^4/8].

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.

Cf. A000290, A000583, A343326, A343368, A343384, A343391, A343397, A343411.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/2]-Floor[y^4/8]],r=r+1],{x,1,Sqrt[2n+1]},{y,1,(8(n-Floor[x^2/2])+7)^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]

A343391 Number of ways to write n as x^2 + [y^2/4] + [z^4/6] with x,y,z positive integers, where [.] is the floor function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 13 2021

nonn

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..10^6.
See also A343387 for a similar conjecture.

			a(1) = 1 with 1 = 1^2 + [1^2/4] + [1^4/6].
a(2) = 1 with 2 = 1^2 + [2^2/4] + [1^4/6].
a(14) = 1 with 14 = 1^2 + [1^2/4] + [3^4/6].
a(32) = 1 with 32 = 4^2 + [8^2/4] + [1^4/6].
a(35) = 1 with 35 = 4^2 + [5^2/4] + [3^4/6].
a(77) = 1 with 77 = 8^2 + [1^2/4] + [3^4/6].
a(840) = 1 with 840 = 28^2 + [15^2/4] + [1^4/6].

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.

Cf. A000290, A000583, A343326, A343368, A343384, A343387, A343397, A343411.

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/4]-Floor[y^4/6]],r=r+1],{x,1,Sqrt[4n+3]},{y,1,(6(n-Floor[x^2/4])+5)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343411 Number of ways to write n as x^3 + [y^3/2] + [z^3/3] + 2^w, where [.] is the floor function, x,y,z are positive integers, and w is a nonnegative integer.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 3, 2, 3, 4, 5, 2, 7, 2, 2, 2, 5, 4, 5, 5, 3, 3, 3, 3, 7, 6, 3, 5, 5, 6, 2, 11, 3, 6, 2, 6, 6, 8, 10, 2, 9, 2, 5, 5, 10, 5, 2, 6, 4, 4, 7, 5, 7, 2, 2, 4, 6, 7, 3, 12, 3, 7, 4, 9, 6, 5, 10, 4, 15, 4, 8, 5, 11, 4, 8, 14, 6, 4, 6, 10, 7, 8, 9, 5, 6, 4, 4, 13, 5, 7, 3, 10, 2, 7, 11

Offset: 1

Zhi-Wei Sun, Apr 14 2021

nonn

Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for all 1 < n <= 3*10^5.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 14 2021

			a(2) = 1 with 2 = 1^3 + [1^3/2] + [1^3/3] + 2^0.
a(3) = 1 with 3 = 1^3 + [1^3/2] + [1^3/3] + 2^1.
a(4) = 1 with 4 = 1^3 + [1^3/2] + [2^3/3] + 2^0.
a(6) = 1 with 6 = 1^3 + [2^3/2] + [1^3/3] + 2^0.
a(8) = 1 with 8 = 1^3 + [2^3/2] + [2^3/3] + 2^0.
a(10) = 1 with 10 = 2^3 + [1^3/2] + [1^3/3] + 2^1.
a(103) = 1 with 103 = 3^3 + [1^3/2] + [6^3/3] + 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.

Cf. A000079, A000578, A343326, A343368, A343384, A343387, A343391, A343397.

CQ[n_]:=CQ[n]=n>0&&IntegerQ[n^(1/3)];
tab={};Do[r=0;Do[If[CQ[n-Floor[x^3/2]-Floor[y^3/3]-2^z],r=r+1],{x,1,(2n-1)^(1/3)},{y,1,(3(n-Floor[x^3/2])-1)^(1/3)},{z,0,Log[2,n-Floor[x^3/2]-Floor[y^3/3]]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343460 Number of ways to write n as x^6 + y^3 + z(3z+1)/2 + 2^k, where x and y are nonnegative integers, z is an integer and k is a positive integer.

Original entry on oeis.org

0, 1, 3, 5, 6, 5, 4, 4, 6, 9, 8, 6, 5, 5, 6, 7, 11, 11, 7, 5, 5, 5, 5, 8, 8, 5, 4, 5, 7, 7, 10, 11, 7, 8, 8, 8, 8, 9, 10, 8, 6, 7, 10, 10, 10, 7, 6, 7, 4, 5, 7, 6, 5, 4, 7, 8, 6, 5, 7, 8, 7, 6, 3, 5, 8, 12, 15, 13, 12, 10, 9, 11, 17, 18, 13, 9, 6, 9, 11, 16

Offset: 1

Zhi-Wei Sun, Apr 16 2021

nonn

Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for n = 2..10^7.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 18 2021

			a(2) = 1 with 2 = 0^6 + 0^3 + 0*(3*0+1)/2 + 2^1.
a(175) = 2 with 175 = 1^6 + 3^3 + (-10)*(3*(-10)+1)/2 + 2^1 = 2^6 + 4^3 + 3*(3*3+1)/2 + 2^5.
a(14553) = 1 with 14553 = 2^6 + 17^3 + (-80)*(3*(-80)+1)/2 + 2^4.

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

Cf. A000079, A000578, A001014, A001318, A343397, A343411.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
tab={};Do[r=0;Do[If[PenQ[n-x^6-y^3-2^k],r=r+1],{x,0,(n-1)^(1/6)},{y,0,(n-x^6-1)^(1/3)},{k,1,Log[2,n-x^6-y^3]}];tab=Append[tab,r],{n,1,80}];Print[tab]

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.

Original entry on oeis.org

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]

A338687 Number of ways to write n as x^4 + y^2 + floor(z^2/7), where x,y,z are integers with x >= 0, y >= 1 and z >= 2.

Original entry on oeis.org

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

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..5*10^6.
See also A338686 for a similar conjecture.

			a(1) = 1 with 1 = 0^4 + 1^2 + floor(2^2/7).
a(75) = 1 with 75 = 0^4 + 8^2 + floor(9^2/7).
a(1799) = 1 with 1799 = 5^4 + 25^2 + floor(62^2/7).
a(7224) = 1 with 7224 = 9^4 + 19^2 + floor(46^2/7).
a(27455) = 2 with 27455 = 0^4 + 7^2 + floor(438^2/7) = 8^4 + 118^2 + floor(257^2/7).

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

Cf. A000290, A000583, A270566, A338686, A338696, A343387, A343391, A343397.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-x^4-Floor[y^2/7]],r=r+1],{x,0,(n-1)^(1/4)},{y,2,Sqrt[7(n-x^4)-1]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343528 Number of ways to write n as x^4 + T(y)^2 + T(z) + 2^w, where x,y,z are nonnegative integers, w is a positive integer, and T(m) denotes the triangular number m*(m+1)/2.

Original entry on oeis.org

0, 1, 3, 4, 5, 5, 3, 4, 6, 5, 5, 6, 5, 6, 6, 4, 7, 10, 10, 9, 7, 4, 7, 10, 7, 8, 9, 7, 5, 7, 7, 10, 13, 9, 8, 7, 5, 8, 14, 9, 10, 11, 6, 9, 10, 8, 10, 13, 8, 7, 6, 5, 11, 15, 9, 7, 8, 6, 8, 10, 10, 10, 10, 6, 7, 9, 6, 10, 17, 10, 9, 9, 6, 10, 10, 6, 9, 9, 6, 10, 9, 6, 11, 14, 8, 11, 11, 9, 11, 11

Offset: 1

Zhi-Wei Sun, Apr 18 2021

nonn

Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for all 1 < n <= 2*10^7.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 21 2021

			a(2) = 1 with 2 = 0^4 + T(0)^2 + T(0) + 2^1.
a(7) = 3, and 7 = 0^4 + T(0)^2 + T(2) + 2^2 = 1^4 + T(1)^2 + T(1) + 2^2 = 1^4 + T(1)^2 + T(2) + 2^1.

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

Cf. A000079, A000217, A000290, A000583, A343397, A343411, A343460.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-x^4-(y(y+1)/2)^2-2^k],r=r+1],{x,0,(n-1)^(1/4)},{y,0,(Sqrt[8*Sqrt[n-x^4-1]+1]-1)/2},{k,1,Log[2,n-x^4-(y(y+1)/2)^2]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A343400 Number of ways to write n as 3^x + [y^2/3] + [z^2/4], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 4, 6, 4, 9, 5, 7, 8, 6, 9, 6, 7, 9, 7, 6, 9, 7, 8, 7, 7, 10, 6, 9, 11, 9, 12, 8, 9, 14, 5, 13, 11, 8, 11, 11, 7, 13, 9, 12, 11, 9, 9, 11, 8, 12, 11, 11, 11, 6, 16, 4, 11, 12, 11, 13, 12, 6, 10, 9, 8, 17, 8, 12, 11, 10, 8, 10, 12, 10, 8, 11, 12, 12, 13, 7

Offset: 1

Zhi-Wei Sun, Apr 13 2021

nonn

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..2*10^6.
The first indices n for which a(n) = 0 are 4051736, 7479656, 8592680, 9712160, 14039792, 16726256, 24914510. - Giovanni Resta, Apr 14 2021

			a(2) = 2 with 2 = 3^0 + [1^2/3] + [2^2/4] = 3^0 + [2^2/3] + [1^2/4].
a(2942) = 2 with 2942 = 3^1 + [93^2/3] + [15^2/4] = 3^7 + [44^2/3] + [21^2/4].
a(627662) = 5 with 627662 - 3^0 = [330^2/3] + [1538^2/4] = [1042^2/3] + [1031^2/4] = [1318^2/3] + [441^2/4] = [1328^2/3] + [399^2/4] = [1352^2/3] + [271^2/4].
a(1103096) = 3 with 1103096 = 3^1 + [260^2/3] + [2079^2/4] = 3^1 + [508^2/3] + [2017^2/4] = 3^9 + [328^2/3] + [2047^2/4].
a(1694294) = 3 with 1694294 = 3^8 + [860^2/3] + [2401^2/4] = 3^8 + [928^2/3] + [2367^2/4] = 3^13 + [112^2/3] + [619^2/4].

Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.

Cf. A000290, A000244, A343387, A343391, A343397.

PowQ[n_]:=PowQ[n]=IntegerQ[Log[3,n]];
tab={};Do[r=0;Do[If[PowQ[n-Floor[x^2/3]-Floor[y^2/4]],r=r+1],{x,1,Sqrt[3n-1]},{y,1,Sqrt[4(n-Floor[x^2/3]-1)+1]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A343503 Number of ways to write n as x(3x+1)/2 + y(7y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 5, 5, 6, 4, 4, 5, 6, 4, 4, 8, 9, 6, 9, 8, 8, 6, 8, 7, 2, 7, 6, 6, 5, 7, 9, 8, 7, 10, 6, 11, 9, 9, 10, 6, 10, 9, 10, 6, 7, 10, 10, 6, 7, 6, 7, 7, 6, 7, 6, 11, 10, 9, 9, 9, 10, 10, 10, 9, 7, 7, 14, 8, 11, 9, 13, 11, 7, 13, 9, 7, 10, 8, 6, 7, 10, 11, 4, 9, 8, 12, 8, 11, 12, 6, 12, 11, 12, 13, 7, 12, 10, 11, 11, 9

Offset: 1

Zhi-Wei Sun, Apr 17 2021

nonn

The author noted that a(n) > 0 for all n = 1..2*10^7. Giovanni Resta found that 8558169401 is the first value of n with a(n) = 0.

			a(1) = 1 with 1 = 0*(3*0+1)/2 + 0*(7*0+1)/2 + 2^0.
a(25) = 2, and 25 = 1*(3*1+1)/2 + 2*(7*2+1)/2 + 2^3 = (-2)*(3*(-2)+1)/2 + 1*(7*1+1)/2 + 2^4.

Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math. 63 (2020), 501-520.

Cf. A000079, A001318, A270566, A343397, A343411, A343460.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
tab={};Do[r=0;Do[If[PenQ[n-2^k-x(7x+1)/2],r=r+1],{k,0,Log[2,n]},{x,-Floor[(Sqrt[56(n-2^k)+1]+1)/14],(Sqrt[56(n-2^k)+1]-1)/14}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A343387 Number of ways to write n as x^2 + [y^2/2] + [z^4/8], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

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A343391 Number of ways to write n as x^2 + [y^2/4] + [z^4/6] with x,y,z positive integers, where [.] is the floor function.

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A343411 Number of ways to write n as x^3 + [y^3/2] + [z^3/3] + 2^w, where [.] is the floor function, x,y,z are positive integers, and w is a nonnegative integer.

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A343460 Number of ways to write n as x^6 + y^3 + z*(3*z+1)/2 + 2^k, where x and y are nonnegative integers, z is an integer and k is a positive integer.

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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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A338687 Number of ways to write n as x^4 + y^2 + floor(z^2/7), where x,y,z are integers with x >= 0, y >= 1 and z >= 2.

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A343528 Number of ways to write n as x^4 + T(y)^2 + T(z) + 2^w, where x,y,z are nonnegative integers, w is a positive integer, and T(m) denotes the triangular number m*(m+1)/2.

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A343400 Number of ways to write n as 3^x + [y^2/3] + [z^2/4], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

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A343460 Number of ways to write n as x^6 + y^3 + z(3z+1)/2 + 2^k, where x and y are nonnegative integers, z is an integer and k is a positive integer.

A343503 Number of ways to write n as x(3x+1)/2 + y(7y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.