A349945 -id:A349945 - cp's OEIS frontend

A349957 Number of ways to write n as x^4 + y^2 + (z^2 + 11*16^w)/60, where x,y,z are nonnegative integers, and w is 0 or 1.

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

Offset: 1

Zhi-Wei Sun, Dec 06 2021

nonn

Conjecture 1: a(n) > 0 for all n > 0.
This has been verified for n <= 10^6. It seems that a(n) = 1 only for n = 1, 71, 78, 247, 542, 1258, 1907, 5225, 19798.
Conjecture 2: (i) If a is 1 or 3, then each n = 0,1,2,... can be written as a*x^8 + y^2 + (7*z^4 + w^2)/64 with x,y,z,w nonnegative integers.
(ii) If a is among 1,2,5, then each n = 0,1,2,... can be written as a*x^8 + y^2 + (11*z^4 + w^2)/60 with x,y,z,w nonnegative integers.
Conjecture 3: If (a,b, m) is among the triples (1,7,8), (1,11,12), (2,7,8), (3,11,12), (5,7,8), then each n = 0,1,2,... can be written as a*x^4 + y^2 + (b*z^6 + w^2)/m with x,y,z,w nonnegative integers.
Conjecture 4: (i) If F(x,y,z,w) is x^6 + y^2 + (5*z^4 + 3*w^2)/16 or 3x^6 + 2*y^2 + (11*z^4 + w^2)/60, then each n = 0,1,2,... can be written as F(x,y,z,w) with x,y,z,w nonnegative integers.
(ii) If (a,b,m) is among the triples (1,7,64), (1,11,12), (1,11,60), (2,1,25), (2,1,65), (2,11,4), (2,11,20), (2,11,60), (4,2,9), (4,7,64), (5,11,60), (6,1,10), then each n = 0,1,2,... can be written as a*x^6 + y^2 + (b*z^4 + w^2)/m with x,y,z,w nonnegative integers.

			a(1) = 1 with 1 = 0^4 + 0^2 + (7^2 + 11*16^0)/60.
a(16) = 2 with 16 = 0^4 + 0^2 + (28^2 + 11*16)/60 = 1^4 + 2^2 + (22^2 + 11*16)/60.
a(71) = 1 with 71 = 0^4 + 2^2 + (62^2 + 11*16)/60.
a(78) = 1 with 78 = 2^4 + 5^2 + (47^2 + 11*16^0)/60.
a(247) = 1 with 247 = 3^4 + 3^2 + (97^2 + 11*16^0)/60.
a(542) = 1 with 542 = 3^4 + 21^2 + (32^2 + 11*16)/60.
a(1258) = 1 with 1258 = 2^4 + 15^2 + (247^2 + 11*16^0)/60.
a(1907) = 1 with 1907 = 0^4 + 0^2 + (338^2 + 11*16)/60.
a(5225) = 1 with 5225 = 5^4 + 58^2 + (272^2 + 11*16)/60.
a(19798) = 1 with 19798 = 1^4 + 137^2 + (248^2 + 11*16)/60.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167--190.
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Cf. A000290, A000583, A001014, A001016, A349942, A349945, A349956.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[60(n-x^4-y^2)-11*16^z],r=r+1],{x,0,(n-1)^(1/4)},{y,0,Sqrt[n-1-x^4]},{z,0,1}];tab=Append[tab,r],{n,1,100}];Print[tab]

A349956 Number of ways to write n as x^2 + 2y^2 + (z^4 + 4w^4)/5 with x,y,z,w nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Dec 06 2021

nonn

a(n) > 0 for all n <= 10^5.
Conjecture: If m is 5 or 65 or 85, then each n = 0,1,2,... can be written as x^2 + 2*y^2 + (z^4 + 4*w^4)/m with x,y,z,w nonnegative integers.
It seems that there are infinitely many positive squarefree numbers m (including 3, 5, 15, 23, 31, 33, 37, 55, 59, 67, 69, 71, 89, 93, 97, 111, 113, 115) such that every n = 0,1,2,... can be written as x^4 + 2*y^4 + (z^2 + 11*w^2)/m with x,y,z,w nonnegative integers.

			a(11) = 1 with 11 = 3^2 + 2*1^2 + (0^4 + 4*0^4)/5.
a(14) = 1 with 14 = 1^2 + 2*0^2 + (1^4 + 4*2^4)/5.
a(78) = 1 with 78 = 7^2 + 2*0^2 + (3^4 + 4*2^4)/5.
a(155) = 1 with 155 = 11^2 + 2*3^2 + (2^4 + 4*2^4)/5.
a(174) = 1 with 174 = 7^2 + 2*0^2 + (5^4 + 4*0^4)/5.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167--190.
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Cf. A000290, A000583, A349942, A349945.

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

cp's OEIS Frontend

A349957 Number of ways to write n as x^4 + y^2 + (z^2 + 11*16^w)/60, where x,y,z are nonnegative integers, and w is 0 or 1.

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A349956 Number of ways to write n as x^2 + 2y^2 + (z^4 + 4w^4)/5 with x,y,z,w nonnegative integers.

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cp's OEIS Frontend

A349957 Number of ways to write n as x^4 + y^2 + (z^2 + 11*16^w)/60, where x,y,z are nonnegative integers, and w is 0 or 1.

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A349956 Number of ways to write n as x^2 + 2*y^2 + (z^4 + 4*w^4)/5 with x,y,z,w nonnegative integers.

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Keywords

Comments

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Mathematica

A349956 Number of ways to write n as x^2 + 2y^2 + (z^4 + 4w^4)/5 with x,y,z,w nonnegative integers.