A347865 -id:A347865 - cp's OEIS frontend

A350857 Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square.

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

Offset: 0

Zhi-Wei Sun, Jan 19 2022

nonn

Conjecture 1: a(n) > 0 except for n = 95, 255.
This has been verified for n up to 10^7.
It seems that a(n) = 1 only for n = 0, 14, 15, 24, 30, 60, 78, 79, 111, 159, 174, 249, 270, 303, 318, 334, 382, 399, 414, 462, 472, 831, 894, 975, 1080, 1151, 1164, 1919, 4080, 6456, 20319.
Conjecture 2: For any positive integer a == 2 (mod 4), all sufficiently large integers can be written as a*w^4 + x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If N(a) denotes the largest integer not of the form a*w^4 + x^4 + (2*y)^2 + z^2 (w,x,y,z = 0,1,2,...), then we have N(2) = 255, N(6) = 2716, N(10) = 598, N(14) = 8427, N(18) = 2463, N(22) = 3884, N(26) = 14988, N(30) = 10843.
See also A346643 and A347865 for similar conjectures.

			a(24) = 1 with 24 = 2*0^4 + 2^4 + 2^2 + 2^2 and 0 = 0^2.
a(1151) = 1 with 1151 = 2*3^4 + 4^4 + 2^2 + 27^2 and 4 = 2^2.
a(6456) = 1 with 6456 = 2*1^4 + 3^4 + 17^2 + 78^2 and 1 = 1^2.
a(20319) = 1 with 20319 = 2*5^4 + 0^4 + 5^2 + 138^2 and 0 = 0^2.

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, A346643, A347865.

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

A346643 Number of ways to write n as w^2 + 2x^2 + 3y^4 + 4*z^4, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Jan 24 2022

nonn

1-2-3-4 Conjecture: a(n) > 0 except for n = 158.
This has been verified for n up to 10^8.
It seems that a(n) = 1 only for n = 0, 1, 2, 14, 17, 35, 42, 46, 62, 87, 119, 122, 168, 189, 206, 234, 237, 302, 317, 398, 545, 1037, 1437, 4254.
See also A347865 and A350857 for similar conjectures.

			a(46) = 1 with 46 = 5^2 + 2*3^2 + 3*1^4 + 4*0^4.
a(119) = 1 with 119 = 7^2 + 2*3^2 + 3*2^4 + 4*1^4.
a(398) = 1 with 398 = 13^2 + 2*9^2 + 3*1^4 + 4*2^4.
a(545) = 1 with 545 = 19^2 + 2*6^2 + 3*2^4 + 4*2^4.
a(1037) = 1 with 1037 = 31^2 + 2*6^2 + 3*0^4 + 4*1^4.
a(1437) = 1 with 1437 = 9^2 + 2*26^2 + 3*0^4 + 4*1^4.
a(4254) = 1 with 4254 = 45^2 + 2*31^2 + 3*3^4 + 4*2^4.

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, A347865, A350857.

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

A214891 Numbers that are not the sum of two squares and two fourth powers.

Original entry on oeis.org

23, 44, 71, 79, 184, 368, 519, 599, 704, 1136, 1264, 2944, 4024, 5888, 8304, 9584, 11264, 18176, 20224, 47104, 64384, 94208, 132864, 153344, 180224, 290816, 323584, 753664, 1030144, 1507328, 2125824, 2453504, 2883584, 4653056, 5177344, 12058624, 16482304

Offset: 1

Joerg Arndt, Jul 29 2012

nonn

From XU Pingya, Feb 07 2018: (Start)
When n is a term, 16n is also. This can be proved as follows:
(1) If w is odd, then 16n - w^4 == 7 (mod 8), and it follows from Legendre's three-square theorem that the equation x^2 + y^2 + z^4 + w^4 = 16n has no solution (it is the same when x, y or z are odd numbers).
(2) If x, y, z and w are even numbers (x = 2a, y = 2b, z = 2c, w = 2d) such that x^2 + y^2 + z^4 + w^4 = 16n, then a^2 + b^2 = 4(n - c^4 - d^4). So there are integers u and v satisfying u^2 + v^2 = n - c^4 - d^4. i.e. u^2 + v^2 + c^4 + d^4 = n, which is a contradiction.
(End)
Conjecture: The set {a(n): n > 0} coincides with {16^k*m: k = 0, 1, 2, ... and m = 23, 44, 71, 79, 184, 519, 599, 4024}. - Zhi-Wei Sun, Jan 27 2022

Donovan Johnson, Table of n, a(n) for n = 1..52 (terms <= 4*10^9)
Zhi-Wei Sun, On w^4+x^4+y^2+z^2 over a number field, Question 414791 at MathOverflow, Jan. 27, 2022.

Cf. A001481, A004999, A022549, A346643, A347865, A350857, A350860.

N=10^6;  x='x+O('x^N);
S(e)=sum(j=0, ceil(N^(1/e)), x^(j^e));
v=Vec( S(4)^2 * S(2)^2 );
for(n=1,#v,if(!v[n],print1(n-1,", ")));

a(29)-a(37) from Donovan Johnson, Jul 29 2012

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.

Original entry on oeis.org

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]

cp's OEIS Frontend

A350857 Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square.

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A346643 Number of ways to write n as w^2 + 2*x^2 + 3*y^4 + 4*z^4, where w,x,y,z are nonnegative integers.

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A214891 Numbers that are not the sum of two squares and two fourth powers.

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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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A346643 Number of ways to write n as w^2 + 2x^2 + 3y^4 + 4*z^4, where w,x,y,z are nonnegative integers.