A214891 -id:A214891 - cp's OEIS frontend

A348890 Number of ways to write n as 5*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with y <= z.

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

Offset: 0

Zhi-Wei Sun, Jan 28 2022

nonn

Conjecture: a(n) = 0 only for n == 12 (mod 16).
This has been verified for n up to 10^8.
Now we show that a(n) = 0 whenever n == 12 (mod 16). If 16*q + 12 = 5*w^4 + x^4 + y^2 + z^2 with q,w,x,y,z integers, then the equality modulo 8 yields that w,x,y,z are all even, hence 4*q + 3 == 20*(w/2)^4 + 4*(x/2)^4 + (y/2)^2 + (z/2)^2 and thus (y/2)^2 + (z/2)^2 == 3 (mod 4) which is impossible.
It seems that a(n) = 1 only for n = 0, 3, 4, 27, 43, 48, 49, 63, 67, 72, 75, 192, 215, 303, 1092.

			a(192) = 1 with 192 = 5*1^4 + 3^4 + 5^2 + 9^2.
a(215) = 1 with 215 = 5*1^4 + 2^4 + 5^2 + 13^2.
a(303) = 1 with 303 = 5*1^4 + 0^4 + 3^2 + 17^2.
a(1092) = 1 with 1092 = 5*0^4 + 2^4 + 20^2 + 26^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000290, A000583, A214891, A346643, A347865, A350857.

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

A351206 Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2022

nonn

Conjecture: a(n) exists for any nonnegative integer n.

This implies that each nonnegative rational number can be written as 7*w^2 + x^4 + y^4 + z^4 with w,x,y,z rational numbers.

			a(6) = 2 with 6 = 1^4 + (1^4 + 2^4 + 7*3^2)/2^4.
a(19) = 6 with 19 = 0^4 + (1^4 + 4^4 + 7*59^2)/6^4.
a(22) = 10 with 22 = 2^4 + (2^4 + 13^4 + 7*67^2)/10^4.
a(5797) = 20 with 5797 = 0^4 + (81^4 + 164^4 + 7*4797^2)/20^4.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000290, A000583, A214891, A348890, A346643, A347827, A347865, A349942, A349943, A350714, A350857, A350860.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[m=1; Label[bb]; k=m^4; Do[If[SQ[(k*(n-x^4)-y^4-z^4)/7], tab=Append[tab,m]; Goto[aa]],  {x, 0, n^(1/4)}, {y, 0, (k*(n-x^4)/2)^(1/4)},{z,y,(k*(n-x^4)-y^4)^(1/4)}]; m=m+1; Goto[bb]; Label[aa], {n,0,100}];Print[tab]

A305553 Numbers that are not the sum of 2 squares and a 4th power.

Original entry on oeis.org

7, 12, 15, 22, 23, 28, 31, 39, 43, 44, 47, 55, 60, 63, 67, 70, 71, 76, 78, 79, 87, 92, 93, 95, 103, 108, 111, 112, 119, 124, 127, 135, 140, 143, 151, 156, 159, 167, 168, 172, 175, 177, 183, 184, 188, 191, 192, 199, 204, 207, 214, 215, 220, 223, 231, 236

Offset: 1

XU Pingya, Jun 20 2018

nonn

Numbers of the form 4*A017101(k) are terms of this sequence.

m is a term iff 16m is also.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.

Subsequence of A000037, A140823 and A022544.
A004215 and A214891 are subsequences.
Cf. A017101, A022552.

n=239;
t=Union@Flatten@Table[x^2+y^2+z^4, {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/4)}];
Complement[Range[0,n], t]

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A348890 Number of ways to write n as 5*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with y <= z.

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A351206 Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.

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A305553 Numbers that are not the sum of 2 squares and a 4th power.

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Mathematica