A347824 -id:A347824 - cp's OEIS frontend

A347827 Number of ways to write n as w^4 + x^4 + (y^2 + 23*z^2)/16, where w is zero or a power of two (including 2^0 = 1), and x,y,z are nonnegative integers.

1, 3, 4, 4, 4, 3, 2, 2, 2, 4, 5, 2, 2, 5, 4, 1, 4, 8, 8, 8, 6, 2, 2, 3, 6, 12, 9, 5, 9, 9, 4, 2, 8, 8, 6, 5, 4, 6, 4, 4, 11, 11, 6, 7, 6, 3, 3, 5, 11, 8, 7, 3, 9, 10, 5, 11, 9, 3, 4, 5, 3, 2, 3, 7, 10, 10, 6, 2, 7, 5, 8, 10, 5, 9, 7, 6, 4, 1, 6, 9, 9, 9, 10, 7, 5, 4, 6, 5, 13, 11, 6, 5, 3, 6, 16, 11, 6, 15, 15, 7, 5

Offset: 0

Zhi-Wei Sun, Jan 23 2022

nonn

23-Conjecture: a(n) > 0 for all n = 0,1,2,....
This is stronger than the conjecture in A347824, and it has been verified for n up to 3*10^6. See also a similar conjecture in A347562.
It seems that a(n) = 1 only for n = 0, 15, 77, 231, 291, 437, 471, 1161, 1402, 4692, 7107, 9727.

			a(231) = 1 with 231 = 0^4 + 3^4 + (10^2 + 23*10^2)/16.
a(437) = 1 with 437 = 3^4 + 4^4 + (40^2 + 23*0^2)/16.
a(1402) = 1 with 1402 = 2^4 + 5^4 + (3^2 + 23*23^2)/16.
a(9727) = 1 with 9727 = 0^4 + 6^4 + (367^2 + 23*3^2)/16.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000079, A000290, A000583, A347562, A347824.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];tab={};Do[r=0;Do[If[w==0||IntegerQ[Log[2,w]],Do[If[SQ[16(n-w^4-x^4)-23z^2],r=r+1],{x,0,(n-w^4)^(1/4)},{z,0,Sqrt[16(n-w^4-x^4)/23]}]],{w,0,n^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]

A347562 Number of ways to write n as 16^w + x^2 + (y^2 + 23*z^4)/324, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 23 2022

nonn

Conjecture: a(n) > 0 for all n > 0.
This has been verified for n up to 10^6.
It seems that a(n) = 1 only for n = 1, 4, 8, 12, 15, 40, 55, 70, 76, 85, 92, 104, 135, 156, 177, 192, 204, 207, 231, 279, 300, 447, 567, 1427, 1887, 4371.
See also A347827 for a similar conjecture.

			a(15) = 1 with 15 = 16^0 + 1^2 + (62^2 + 23*2^4)/324.
a(156) = 1 with 156 = 16^1 + 6^2 + (139^2 + 23*5^4)/324.
a(300) = 1 with 300 = 16^2 + 6^2 + (27^2 + 23*3^4)/324.
a(1427) = 1 with 1427 = 16^1 + 35^2 + (71^2 + 23*7^4)/324.
a(1887) = 1 with 1887 = 16^1 + 15^2 + (729^2 + 23*3^4)/324.
a(4371) = 1 with 4371 = 16^1 + 63^2 + (351^2 + 23*3^4)/324.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000290, A000583, A001025, A347824, A347827, A350860.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[324(n-16^w-x^2)-23y^4],r=r+1],{w,0,Log[16,n]},{x,0,Sqrt[n-16^w]},
{y,0,(324(n-16^w-x^2)/23)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A347827 Number of ways to write n as w^4 + x^4 + (y^2 + 23*z^2)/16, where w is zero or a power of two (including 2^0 = 1), and x,y,z are nonnegative integers.

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