A230747 -id:A230747 - cp's OEIS frontend

A340274 Number of ways to write n as x + y + z with x, y, z positive integers such that 3x^2y^2 + 5y^2z^2 + 8z^2x^2 is a square.

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 4, 2, 5, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 3, 3, 4, 5, 2, 3, 5, 5, 4, 4, 2, 4, 4, 5, 3, 4, 3, 6, 3, 2, 5, 2, 2, 7, 7, 1, 3, 6, 4, 4, 3, 3, 6, 2, 5, 5, 3, 6, 5, 4, 6, 6, 6, 3, 6, 6, 4, 5, 6, 2, 6, 3, 5, 4, 5, 3, 5, 12, 4, 4, 5, 1, 6, 6, 7, 9, 3, 3, 6, 5, 6, 7, 4

Offset: 1

Zhi-Wei Sun, Apr 24 2021

nonn

Conjecture 1: a(n) > 0 for all n > 2.
We have verified a(n) > 0 for all n = 3..10000. The conjecture holds if a(p) > 0 for every odd prime p. For any n > 0 we have a(3*n) > 0, since 3*n = n + n + n and 3 + 5 + 8 = 4^2.
It seems that a(n) = 1 only for n = 3..8, 10, 11, 19, 53, 89, 127, 178, 257, 461.
See also A343862 for similar conjectures.
Conjecture 1 holds for all n < 2^15. Note a(1823) = 1. - Martin Ehrenstein, May 03 2021

			a(4) = 1 with 4 = 2 + 1 + 1 and 3*2^2*1^2 + 5*1^2*1^2 + 8*1^2*2^2 = 7^2.
a(19) = 1 with 19 = 9 + 9 + 1 and 3*9^2*9^2 + 5*9^2*1^2 + 8*1^2*9^2 = 144^2.
a(53) = 1 with 53 = 23 + 7 + 23 and 3*23^2*7^2 + 5*7^2*23^2 + 8*23^2*23^2 = 1564^2.
a(89) = 1 with 89 = 2 + 58 + 29 and 3*2^2*58^2 + 5*58^2*29^2 + 8*29^2*2^2 = 3770^2.
a(257) = 1 with 257 = 11 + 164 + 82 and 3*11^2*164^2 + 5*164^2*82^2 + 8*82^2*11^2 = 30340^2.
a(461) = 1 with 461 = 186 + 165 + 110 and 3*186^2*165^2 + 5*165^2*110^2 + 8*110^2*186^2 = 88440^2.

Martin Ehrenstein, Table of n, a(n) for n = 1..32767 (first 1500 terms from Zhi-Wei Sun)
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175 (2017), 167-190. See also arXiv version, arXiv:1604.06723 [math.NT], 2016-2017.

Cf. A000041, A000290, A230121, A230747, A231168, A262357, A271719, A273278, A343862.

SQ[n_]:=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[3x^2*y^2+(n-x-y)^2*(5*y^2+8*x^2)],r=r+1],{x,1,n-2},{y,1,n-1-x}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343862 Number of ways to write n as x + y + z with x, y, z positive integers such that x^2y^2 + 5y^2z^2 + 10z^2*x^2 is a square.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 1, 1, 4, 2, 4, 2, 2, 5, 5, 3, 3, 2, 6, 4, 3, 3, 6, 6, 5, 2, 6, 4, 10, 3, 6, 4, 6, 6, 8, 5, 6, 4, 9, 7, 6, 3, 7, 9, 5, 5, 15, 5, 12, 11, 10, 5, 6, 7, 10, 8, 9, 7, 15, 7, 6, 7, 10, 10, 7, 9, 10, 10, 12, 4, 15, 9, 9, 11, 9, 7, 12, 11, 15, 8, 9, 7, 12, 10, 3, 9, 11, 11, 19, 12, 12, 9, 10, 6, 23, 11, 6, 10, 18

Offset: 1

Zhi-Wei Sun, May 02 2021

nonn

Conjecture 1: a(n) > 0 for all n > 2.
We have verified a(n) > 0 for all n = 3..10000. Conjecture 1 holds if a(p) > 0 for each odd prime p. For any n > 0 we have a(3*n) > 0 since 3*n = n + n + n and 1 + 5 + 10 = 4^2.
See also A340274 for a similar conjecture.
Conjecture 2: There are infinitely many triples (a,b,c) of positive integers such that each n = 3,4,... can be written as x + y + z with x,y,z positive integers and a*x^2*y^2 + b*y^2*z^2 + c*z^2*x^2 a square.
Such triple candidates include (21,19,9), (23,17,9), (24,16,9), (25,14,10), (29,19,16), (33,27,21), (35,9,5), (37,32,31) etc.
Conjecture 1 holds for all n < 2^15. - Martin Ehrenstein, May 02 2021

			a(4) = 1 with 4 = 2 + 1 + 1 and 2^2*1^2 + 5*1^2*1^2 + 10*1^2*2^2 = 7^2.
a(5) = 1 with 5 = 1 + 3 + 1 and 1^2*3^2 + 5*3^2*1^2 + 10*1^1*1^2 = 8^2.
a(8) = 1 with 8 = 4 + 2 + 2 and 4^2*2^2 + 5*2^2*2^2 + 10*2^2*4^2 = 28^2.
a(9) = 1 with 9 = 3 + 3 + 3 and 3^2*3^2 + 5*3^2*3^2 + 10*3^2*3^2 = 36^2.
a(19) = 2. We have 19 = 4 + 5 + 10 with 4^2*5^2 + 5*5^2*10^2 + 10*10^2*4^2 = 170^2, and 19 = 4 + 13 + 2 with 4^2*13^2 + 5*13^2*2^2 + 10*2^2*4^2 = 82^2.

Martin Ehrenstein, Table of n, a(n) for n = 1..32767 (first 1500 terms from Zhi-Wei Sun)
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].

Cf. A000041, A000290, A230121, A230747, A231168, A262357, A271719, A273278, A340274.

SQ[n_]:=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[x^2*y^2+(n-x-y)^2*(5*y^2+10*x^2)],r=r+1],{x,1,n-2},{y,1,n-1-x}];tab=Append[tab,r],{n,1,100}];Print[tab]

A231168 Number of ways to write n = x + y + z (x, y, z > 0) such that x^2 + y^2 + z^2 + z is a square, and 6x + 1, 6y - 1, 6*z -1 are all prime.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 2, 2, 2, 1, 3, 2, 5, 1, 4, 3, 2, 3, 1, 1, 4, 2, 5, 3, 3, 4, 4, 8, 2, 3, 8, 2, 4, 3, 4, 8, 7, 2, 2, 8, 3, 8, 6, 1, 6, 8, 4, 1, 9, 2, 4, 10, 6, 1, 7, 11, 7, 10, 2, 6, 9, 3, 6, 3, 6, 6, 6, 8, 4, 8, 4, 4, 9, 2, 11, 4, 9, 6, 1, 4, 5, 5, 10, 7, 5, 6, 6, 7, 5, 8, 17, 8, 5, 2, 7, 8, 11, 10, 6, 4

Offset: 1

Zhi-Wei Sun, Nov 04 2013

nonn

Conjecture: a(n) > 0 for all n > 5.

			a(19) = 1 since 19 = 13 + 5 + 1 with 13^2 + 5^2 + 1^2 + 1 = 14^2, and 6*13 + 1 = 79, 6*5 - 1 = 29, 6*1 - 1 = 5 are all prime.
a(444) = 1 since 444 = 76 + 28 + 340 with 76^2 + 28^2 + 340^2 + 340 = 350^2, and 6*76 + 1 = 457, 6*28 - 1 = 167, 6*340 - 1 = 2039 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A conjecture involving squares and primes, a message to Number Theory List, Nov. 5, 2013.

Cf. A000040, A000290, A227877, A230121, A230596, A230747.

SQ[n_]:=IntegerQ[Sqrt[n]]
a[n_]:=Sum[If[PrimeQ[6i+1]&&PrimeQ[6j-1]&&PrimeQ[6(n-i-j)-1]&&SQ[i^2+j^2+(n-i-j)^2+(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
Table[a[n],{n,1,100}]

A343897 Number of ways to write n as 2x + y + z with x,y,z positive integers such that 16x^2y^2 + 19y^2z^2 + 29z^2*x^2 is a square.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 4, 2, 1, 3, 1, 3, 1, 5, 2, 2, 6, 1, 1, 5, 5, 6, 2, 2, 4, 2, 4, 5, 6, 3, 2, 3, 2, 5, 2, 7, 10, 4, 1, 3, 3, 10, 9, 2, 5, 5, 10, 6, 7, 6, 7, 8, 8, 4, 7, 4, 5, 8, 2, 4, 4, 13, 9, 5, 6, 10, 11, 6, 11, 6, 6, 5, 4, 4, 10, 9, 8, 8, 8, 8, 9, 16, 5, 5, 6, 4, 7, 3, 12, 7, 11, 13

Offset: 1

Zhi-Wei Sun, May 03 2021

nonn

Conjecture: a(n) > 0 for all n > 3.
We have verified a(n) > 0 for all n = 4..10000. The conjecture holds if a(p) > 0 for each odd prime p.
It seems that a(n) = 1 only for n = 4..9, 13, 17, 19, 21, 26, 27, 47.

			a(4) = 1 with 4 = 2*1 + 1 + 1 and 16*1^2*1^2 + 19*1^2*1^2 + 29*1^2*1^2 = 8^2.
a(6) = 1 with 6 = 2*1 + 2 + 2 and 16*1^2*2^2 + 19*2^2*2^2 + 29*2^2*1^2 = 22^2.
a(9) = 1 with 9 = 2*2 + 4 + 1 and 16*2^2*4^2 + 19*4^2*1^2 + 29*1^2*2^2 = 38^2.
a(13) = 1 with 13 = 2*5 + 2 + 1 and 16*5^2*2^2 + 19*2^2*1^2 + 29*1^2*5^2 = 49^2.
a(19) = 1 with 19 = 2*2 + 14 + 1 and 16*2^2*14^2 + 19*14^2*1^2 + 29*1^2*2^2 = 128^2.
a(47) = 1 with 47 = 2*13 + 13 + 8 and 16*13^2*13^2 + 19*13^2*8^2 + 29*8^2*13^2 = 988^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1500

Cf. A000041, A000290, A230121, A230747, A340274, A343862.

SQ[n_]:=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[16(x*y)^2+(n-2x-y)^2*(19*y^2+29x^2)],r=r+1],{x,1,(n-2)/2},{y,1,n-1-2x}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343950 Number of ways to write n as x + y + z with x^2 + 4y^2 + 5z^2 a square, where x,y,z are positive integers with y or z a positive power of two.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 3, 1, 2, 2, 2, 3, 1, 4, 3, 2, 3, 3, 4, 4, 2, 1, 4, 6, 4, 2, 3, 12, 5, 3, 5, 8, 4, 5, 5, 8, 4, 7, 4, 4, 4, 7, 5, 5, 1, 4, 6, 5, 6, 6, 10, 7, 4, 9, 5, 10, 16, 7, 7, 9, 6, 5, 5, 14, 8, 6, 6, 3, 7, 1, 5, 4, 10, 5, 7, 10, 8, 13, 10, 3, 4, 8, 5, 12, 7, 20, 9, 12, 5, 8, 1, 9, 4, 8, 9, 8, 7, 4, 10

Offset: 1

Zhi-Wei Sun, May 05 2021

nonn

Conjecture 1: a(n) > 0 for all n > 7.
We have verified a(n) > 0 for all n = 8..50000. Clearly, a(2*n) > 0 if a(n) > 0.
Conjecture 2: For any integer n > 7, we can write n as x + y + z with x,y,z positive integers such that x^2 + 2*y^2 + 3*z^2 is a square.
Conjecture 3: For any integer n > 4, we can write n as x + y + z with x,y,z positive integers such that 3*x^2 + 4*y^2 + 5*z^2 (or x^2 + 3*y^2 + 5*z^2) is a square.

			a(4) = 1, and 4 = 1 + 1 + 2 with 1^2 + 4*1^2 + 5*2^2 = 5^2.
a(5) = 1, and 5 = 2 + 2 + 1 with 2^2 + 4*2^2 + 5*1^2 = 5^2.
a(9) = 1, and 9 = 4 + 1 + 4 with 4^2 + 4*1^2 + 5*4^2 = 10^2.
a(14) = 1, and 14 = 7 + 5 + 2 with 7^2 + 4*5^2 + 5*2^2 = 13^2.
a(23) = 1, and 23 = 7 + 8 + 8 with 7^2 + 4*8^2 + 5*8^2 = 25^2.
a(46) = 1, and 46 = 14 + 16 + 16 with 14^2 + 4*16^2 + 5*16^2 = 50^2.
a(71) = 1, and 71 = 42 + 8 + 21 with 42^2 + 4*8^2 + 5*21^2 = 65^2.
a(92) = 1, and 92 = 28 + 32 + 32 with 28^2 + 4*32^2 + 5*32^2 = 100^2.
a(142) = 1, and 142 = 84 + 16 + 42 with 84^2 + 4*16^2 + 5*42^2 = 130^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3200
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].

Cf. A000079, A000290, A271510, A271513, A271518, A230121, A230747.

PowQ[n_]:=PowQ[n]=n>1&&IntegerQ[Log[2,n]];
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(PowQ[y]||PowQ[n-x-y])&&SQ[x^2+4*y^2+5*(n-x-y)^2],r=r+1],{x,1,n-3},{y,1,n-1-x}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.

Original entry on oeis.org

284, 1388, 2139, 4772, 8556, 8971, 10836, 21163, 28847, 45707, 54507, 71292, 73348, 95127, 101503, 104228, 131388, 136148, 263172, 350076, 638164, 982292, 1532148, 1687828, 1705407, 1958924, 2082188, 2299364, 2360347, 2728379, 3202356, 4042799, 5046771, 5165332, 5235323, 5560627, 7191079, 7740547, 8041364

Offset: 1

Zhi-Wei Sun, May 04 2021

nonn

Conjecture: The sequence has infinitely many terms.
It is easy to see that no term is divisible by 5. In the b-file we list all the 62 terms not exceeding 10^8.
Note that 2*(n^2+3)^2 - 2^4 = (n+1)^4 + (n-1)^4 with (n+1) - (n-1) = 2. This implies that any integer n > 4 can be written as  x + y + 2^(z-1) with x,y,z positive integers such that x^4 + y^4 + (2^z)^4 is twice a square.
See also A343913 for a similar conjecture.

			a(1) = 284, and 2*284^2 - 2^4 = 20^4 + 6^4 with |20-6| > 2.
a(62) = 97077407, and 2*97077407^2 - 2^4 = 18848045899687282 = 11563^4 + 5583^4 with |11563-5583| > 2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..62

Cf. A000290, A000583, A003336, A230121, A230747, A340274, A343862, A343897, A343913.

QQ[n_]:=IntegerQ[n^(1/4)];
n=0;Do[Do[If[QQ[2*m^2-16-x^4]&&(2*m^2-16-x^4)^(1/4)-x>2,n=n+1;Print[n," ",m];Goto[aa]],{x,0,(m^2-8)^(1/4)}];Label[aa],{m,3,8041364}]

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A340274 Number of ways to write n as x + y + z with x, y, z positive integers such that 3*x^2*y^2 + 5*y^2*z^2 + 8*z^2*x^2 is a square.

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A343862 Number of ways to write n as x + y + z with x, y, z positive integers such that x^2*y^2 + 5*y^2*z^2 + 10*z^2*x^2 is a square.

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A231168 Number of ways to write n = x + y + z (x, y, z > 0) such that x^2 + y^2 + z^2 + z is a square, and 6*x + 1, 6*y - 1, 6*z -1 are all prime.

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A343897 Number of ways to write n as 2*x + y + z with x,y,z positive integers such that 16*x^2*y^2 + 19*y^2*z^2 + 29*z^2*x^2 is a square.

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A343950 Number of ways to write n as x + y + z with x^2 + 4*y^2 + 5*z^2 a square, where x,y,z are positive integers with y or z a positive power of two.

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A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.

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A340274 Number of ways to write n as x + y + z with x, y, z positive integers such that 3x^2y^2 + 5y^2z^2 + 8z^2x^2 is a square.

A343862 Number of ways to write n as x + y + z with x, y, z positive integers such that x^2y^2 + 5y^2z^2 + 10z^2*x^2 is a square.

A231168 Number of ways to write n = x + y + z (x, y, z > 0) such that x^2 + y^2 + z^2 + z is a square, and 6x + 1, 6y - 1, 6*z -1 are all prime.

A343897 Number of ways to write n as 2x + y + z with x,y,z positive integers such that 16x^2y^2 + 19y^2z^2 + 29z^2*x^2 is a square.

A343950 Number of ways to write n as x + y + z with x^2 + 4y^2 + 5z^2 a square, where x,y,z are positive integers with y or z a positive power of two.