A290491 -id:A290491 - cp's OEIS frontend

A334086 Positive numbers not of the form 2x^4 + y(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers.

19, 82, 109, 118, 145, 149, 271, 280, 296, 349, 350, 371, 392, 454, 491, 643, 670, 692, 754, 755, 923, 937, 986, 989, 1021, 1031, 1150, 1189, 1210, 1294, 1346, 1372, 1610, 1682, 1699, 1720, 1819, 1913, 2050, 2065, 2141, 2227, 2479, 2524, 2753, 2996, 3184, 3451, 3590, 3805, 3968, 4129, 4139, 4199, 4261, 4706

Offset: 1

Zhi-Wei Sun, Apr 14 2020

nonn

Conjecture: The sequence has totally 216 terms as listed in the b-file.
As none of the 216 terms in the b-file is divisible by 3, the conjecture implies that for each nonnegative integer n we can write 3*n as 2*x^4 + y*(y+1)/2 + z*(z+1)/2 and hence 12*n+1 = 8*x^4 + (y+z+1)^2 + (y-z)^2, where x,y,z are integers.
Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8.
It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square.
a(217) > 10^9, if it exists. - Giovanni Resta, Apr 14 2020

			a(1) = 19 since 19 is the first nonnegative integer which cannot be written as the sum of two triangular numbers and twice a fourth power.

Zhi-Wei Sun, Table of n, a(n) for n = 1..216
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (Cf. Conjecture 1.4(ii).)

Cf. A000217, A000583, A115160, A290491, A306227, A334113.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[n-2x^4-y(y+1)/2],Goto[aa]],{x,0,(n/2)^(1/4)},{y,0,(Sqrt[4(n-2x^4)+1]-1)/2}];tab=Append[tab,n];Label[aa],{n,0,5000}];Print[tab]

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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A334086 Positive numbers not of the form 2x^4 + y(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers.

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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

A334086 Positive numbers not of the form 2*x^4 + y*(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers.

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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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A334086 Positive numbers not of the form 2x^4 + y(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers.

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.