A290472 -id:A290472 - cp's OEIS frontend

A290491 Number of ways to write 12n+1 as x^2 + 4y^2 + 8*z^4, where x and y are positive integers and z is a nonnegative integer.

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

Offset: 1

Zhi-Wei Sun, Aug 03 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4, 10, 18, 28, 39, 49, 98, 142, 163, 184, 208, 320, 382, 408, 814, 910, 1414, 2139, 2674, 3188, 3213, 4230, 6279, 25482.
(ii) All the numbers 16*n+5 (n = 0,1,2,...) can be written as x^4 + 4*y^2 + z^2, where x,y,z are integers with y > 0 and z > 0.
(iii) All the numbers 24*n+1 (n = 0,1,2,...) can be written as 12*x^4 + 4*y^2 + z^2 with x,y,z integers. Also, all the numbers 24*n+9 (n = 0,1,2,...) can be written as 2*x^4 + 6*y^2 + z^2 with x,y,z positive integers.
(iv) All the numbers 24*n+2 (n = 0,1,2,...) can be written as x^4 + 9*y^2 + z^2, where x,y,z are integers with z > 0. Also, all the numbers 24*n+17 (n = 0,1,2,...) can be written as x^4 + 16*y^2 + z^2, where x,y,z are integers with y > 0 and z > 0.
(v) All the numbers 30*n+3 (n = 1,2,3,...) can be written as 2*x^4 + 3*y^2 + z^2 with x,y,z positive integers. Also, all the numbers 30*n+21 (n = 0,1,2,...) can be written as 2*x^4 + 3*y^2 + z^2, where x,y,z are integers with z > 0.

			a(10) = 1 since 12*10+1 = 7^2 + 4*4^2 + 8*1^4.
a(28) = 1 since 12*28+1 = 9^2 + 4*8^2 + 8*0^4.
a(49) = 1 since 12*49+1 = 19^2 + 4*5^2 + 8*2^4.
a(3188) = 1 since 12*3188+1 = 103^2 + 4*80^2 + 8*4^4.
a(3213) = 1 since 12*3213+1 = 91^2 + 4*87^2 + 8*0^4.
a(4230) = 1 since 12*4230+1 = 223^2 + 4*16^2 + 8*1^4.
a(6279) = 1 since 12*6279+1 = 19^2 + 4*75^2 + 8*9^4.
a(25482) = 1 since 12*25482+1 = 531^2 + 4*58^2 + 8*6^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), 1367-1396.
Zhi-Wei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 2015-2017.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), No. 2, 97-120.

Cf. A000290, A000583, A270566, A286885, A286944, A287616, A290342, A290472.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[12n+1-8x^4-4y^2],r=r+1],{x,0,((12n+1)/8)^(1/4)},{y,1,Sqrt[(12n+1-8x^4)/4]}];Print[n," ",r],{n,1,100}]

A260418 Number of ways to write 12n+5 as 4x^4 + 4*y^2 + z^2, where x is a nonnegative integer, and y and z are positive integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Aug 04 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4, 54, 159, 289, 999, 1175, 1404, 16391, 39688.
(ii) All the numbers 24*n+4 (n = 0,1,2,...) can be written as 3*x^4+24*y^2+z^2, where x,y,z are integers with x > 0 and z > 0.
(iii) All the numbers 24*n+13 (n = 0,1,2,...) can be written as 3*x^4+9*y^2+z^2 with x,y,z positive integers.
(iv) All the numbers 24*n+r (n = 0,1,2,...) can be written as a*x^4+b*y^2+c*z^2 with x an integer and y and z positive integers, provided that (r,a,b,c) is among the following quadruples: (5,3,1,1), (5,12,4,1), (7,3,6,1), (10,5,1,1), (11,3,8,3), (11,6,3,2), (17,9,4,1).
See A290491 for a similar conjecture.

			a(4) = 1 since 12*4+5 = 4*0^4 + 4*1^2 + 7^2.
a(54) = 1 since 12*54+5 = 4*0^4 + 4*11^2 + 13^2.
a(159) = 1 since 12*159+5 = 4*0^4 + 4*4^2 + 43^2.
a(289) = 1 since 12*289+5 = 4*1^4 + 4*19^2 + 45^2.
a(999) = 1 since 12*999+5 = 4*7^4 + 4*21^2 + 25^2.
a(1175) = 1 since 12*1175+5 = 4*3^4 + 4*55^2 + 41^2.
a(1404) = 1 since 12*1404+5 = 4*3^4 + 4*10^2 + 127^2.
a(16391) = 1 since 12*16391+5 = 4*5^4 + 4*207^2 + 151^2.
a(39688) = 1 since 12*39688+5 = 4*5^4 + 4*50^2 + 681^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), 1367-1396.
Zhi-Wei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 2015-2017.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000290, A000583, A270566, A286885, A286944, A287616, A290342, A290472, A290491.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[12n+5-4x^4-4y^2],r=r+1],{x,0,((12n+5)/4)^(1/4)},{y,1,Sqrt[(12n+5-4x^4)/4]}];Print[n," ",r],{n,0,100}]

A291123 Number of ways to write 4*n+1 as (p-1)^2 + q^2 + r^2, where p is prime and q and r are nonnegative integers with q <= r.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Aug 17 2017

nonn

Conjecture: (i) a(n) > 0 for all n, and a(n) = 1 only for n = 0, 3, 6, 23, 44.
(ii) Any positive integer relatively prime to 6 can be written as (p-1)^2 + q^2 + 3*r^2, where p is prime, and q and r are integers.
(iii) Let n be any nonnegative integer. Then 4*n+1 can be written as x^2 + y^2 + z^2, where x,y,z are nonnegative integers with x + y + 2*z prime; 4*n+2 can be written as x^2 + y^2 + z^2, where x,y,z are nonnegative integers with x + 3*y prime. Also, we can write 4*n+3 as 2*x^2 + y^2 + z^2, where x,y,z are nonnegative integers with 2*x+1 prime.
(iv) Let n be any nonnegative integer. Then we can write 4*n+1 as x^2 + y^2 + z^2 with x,y,z integers such that x^2 + 5*y^2 + 7*z^2 prime, and write 4*n+2 as x^2 + y^2 + z^2 with x,y,z integers such that x^2 + 2*y^2 + 5*z^2 prime. Also, we can write 8*n+3 as x^2 + y^2 + z^2 with x,y,z integers such that 2*x^2 + 4*y^2 + 5*z^2 is prime.
Note that by the Gauss-Legendre theorem any positive integer not of the form 4^k*(8*m+7) (k,m = 0,1,2,...) can be written as the sum of three squares.

			a(0) = 1 since 4*0+1 = (2-1)^2 + 0^2 + 0^2 with 2 prime.
a(3) = 1 since 4*3+1 = (3-1)^2 + 0^2 + 3^2 with 3 prime.
a(6) = 1 since 4*6+1 = (5-1)^2 + 0^2 + 3^2 with 5 prime.
a(23) = 1 since 4*23+1 = (3-1)^2 + 5^2 + 8^2 with 3 prime.
a(44) = 1 since 4*44+1 = (3-1)^2 + 2^2 + 13^2 with 3 prime.

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, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000040, A000290, A260418, A271518, A290472, A290491, A291047.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[PrimeQ[p]&&SQ[4n+1-(p-1)^2-q^2],r=r+1],{p,2,Sqrt[4n+1]+1},{q,0,Sqrt[(4n+1-(p-1)^2)/2]}];Print[n," ",r],{n,0,80}]

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A290491 Number of ways to write 12n+1 as x^2 + 4y^2 + 8*z^4, where x and y are positive integers and z is a nonnegative integer.

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A260418 Number of ways to write 12n+5 as 4x^4 + 4*y^2 + z^2, where x is a nonnegative integer, and y and z are positive integers.

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A291123 Number of ways to write 4*n+1 as (p-1)^2 + q^2 + r^2, where p is prime and q and r are nonnegative integers with q <= r.

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cp's OEIS Frontend

A290491 Number of ways to write 12*n+1 as x^2 + 4*y^2 + 8*z^4, where x and y are positive integers and z is a nonnegative integer.

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A260418 Number of ways to write 12*n+5 as 4*x^4 + 4*y^2 + z^2, where x is a nonnegative integer, and y and z are positive integers.

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A291123 Number of ways to write 4*n+1 as (p-1)^2 + q^2 + r^2, where p is prime and q and r are nonnegative integers with q <= r.

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A290491 Number of ways to write 12n+1 as x^2 + 4y^2 + 8*z^4, where x and y are positive integers and z is a nonnegative integer.

A260418 Number of ways to write 12n+5 as 4x^4 + 4*y^2 + z^2, where x is a nonnegative integer, and y and z are positive integers.