A000378 -id:A000378 - cp's OEIS frontend

A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

9, 17, 18, 25, 26, 27, 29, 33, 34, 36, 38, 45, 49, 51, 53, 57, 59, 61, 62, 68, 69, 72, 73, 75, 77, 82, 83, 85, 94, 97, 100, 102, 104, 105, 106, 107, 108, 109, 116, 118, 123, 130, 132, 136, 138, 139, 141, 144, 147, 152, 154, 155, 157, 158, 165, 177, 180, 187

Offset: 1

Wolfdieter Lang, Apr 08 2013

nonn

These are the numbers for which A000164(a(n)) = 2.
a(n) is the n-th largest number which has a representation as sum of three integer squares (square 0 allowed), in exactly two ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity for a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

			a(1) = 9 = 0^2 + 0^2 + 3^2 = 1^2 + 2^2 + 2^2, and 9 is the smallest number m with A000164(m) = 2 for m >= 0. The multiplicity with order and signs taken into account is 2*3 + 8*3 = 30 = A005875(9).
The two representations [a,b,c] for a(n), n = 1, ..., 10, are
n=1,   9 = [0, 0, 3], [1, 2, 2],
n=2,  17 = [0, 1, 4], [2, 2, 3],
n=3,  18 = [0, 3, 3], [1, 1, 4],
n=4,  25 = [0, 0, 5], [0, 3, 4],
n=5,  26 = [0, 1, 5], [1, 3, 4],
n=6,  27 = [1, 1, 5], [3, 3, 3],
n=7,  29 = [0, 2, 5], [2, 3, 4],
n=8,  33 = [1, 4, 4], [2, 2, 5],
n=9,  34 = [0, 3, 5], [3, 3, 4],
n=10, 36 = [0, 0, 6], [2, 4, 4].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000164, A005875, A000378, A094942 (one way), A224443 (three ways).

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^22, 3, min(3, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
    end:
a:= proc(n) option remember; local k;
      for k from 1 +`if`(n=1, 0, a(n-1))
      while b(k, isqrt(k), 3)<>2 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 09 2013

Select[ Range[0, 200], Length[ PowersRepresentations[#, 3, 2]] == 2 &] (* Jean-François Alcover, Apr 09 2013 *)

This sequence gives the increasingly ordered elements of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly two such representation}.

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 2, m >= 0}.

A047449 Numbers that are primitively represented by x^2 + y^2 + z^2.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 25, 26, 27, 29, 30, 33, 34, 35, 37, 38, 41, 42, 43, 45, 46, 49, 50, 51, 53, 54, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 73, 74, 75, 77, 78, 81, 82, 83, 85, 86, 89, 90, 91, 93, 94, 97, 98, 99, 101, 102, 105, 106

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1)

Cf. A000378, A004215, A034043-A034047.

LinearRecurrence[{1,0,0,0,1,-1},{1,2,3,5,6,9},70] (* Harvey P. Dale, Mar 05 2015 *)

a(n)=(n-1)\5*8+[6,1,2,3,5][n%5+1] \\ Charles R Greathouse IV, Jun 11 2015

Numbers that are congruent to {1, 2, 3, 5, 6} mod 8.
Union of A047449 and A034045 is A000378. Intersection of A047449 and A034043 is A034046. Numbers that are in A000378 and not congruent to 0 mod 4. - Ray Chandler, Sep 05 2004
G.f.: x*(1 + x + x^2 + 2*x^3 + x^4 + 2*x^5) / ( (x^4 + x^3 + x^2 + x + 1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
a(n) = a(n-1) + a(n-5) - a(n-6); a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=6, a(6)=9. - Harvey P. Dale, Mar 05 2015

A034043 Numbers that are imprimitively represented by x^2+y^2+z^2.

Original entry on oeis.org

0, 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 96, 98, 99, 100, 104, 108, 116, 117, 120, 121, 125, 126, 128, 132, 136, 140, 144, 147, 148, 150, 152, 153, 160, 162, 164, 168, 169

Offset: 1

N. J. A. Sloane

nonn

Cf. A000378, A004215, A047449, A034044-A034047.

Union of A034043 and A034044 is A000378. Intersection of A047449 and A034043 is A034046. Numbers that are in A000378 and not squarefree. - Ray Chandler, Sep 05 2004

A034045 Numbers that are imprimitively but not primitively represented by x^2+y^2+z^2.

Original entry on oeis.org

0, 4, 8, 12, 16, 20, 24, 32, 36, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 84, 88, 96, 100, 104, 108, 116, 120, 128, 132, 136, 140, 144, 148, 152, 160, 164, 168, 172, 176, 180, 184, 192, 196, 200, 204, 208, 212, 216, 224, 228, 232, 236, 244, 248, 256

Offset: 1

N. J. A. Sloane

nonn

Cf. A000378, A004215, A047449, A034043-A034047.

Union of A047449 and A034045 is A000378. Union of A034044 and A034045 is A034047. Numbers that are in A000378 and congruent to 0 mod 4. - Ray Chandler, Sep 05 2004

A034047 Numbers that are primitively or imprimitively represented by x^2+y^2+z^2, but not both.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 26, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 48, 51, 52, 53, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 82, 83, 84, 85, 86, 88, 89, 91, 93

Offset: 1

N. J. A. Sloane

nonn

Cf. A000378, A004215, A047449, A034043-A034046.

Union of A034046 and A034047 is A000378. Union of A034044 and A034045 is A034047. Numbers that are in A000378 and either squarefree or congruent to 0 mod 4. - Ray Chandler, Sep 05 2004

A294577 Numbers that are the sum of three squares (square 0 allowed) in exactly four ways.

Original entry on oeis.org

81, 89, 101, 125, 129, 134, 149, 161, 162, 170, 171, 173, 189, 198, 201, 233, 241, 242, 243, 245, 246, 249, 250, 251, 254, 270, 274, 278, 285, 289, 294, 299, 324, 339, 349, 356, 361, 363, 370, 371, 378, 387, 390, 393, 395, 404, 406, 411, 417, 429, 433, 451

Offset: 1

Robert Price, Nov 02 2017

nonn

These are the numbers for which A000164(a(n)) = 4.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly four ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..1030

Cf. A000164, A005875, A000378, A094942, A224442, A224443.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 4 &]

Updated Mathematica program to Version 11. by Robert Price, Nov 01 2019

A292313 Numbers that are the sum of three squares in arithmetic progression.

Original entry on oeis.org

75, 300, 507, 675, 867, 1200, 1875, 2028, 2523, 2700, 3468, 3675, 4107, 4563, 4800, 5043, 6075, 7500, 7803, 8112, 8427, 9075, 10092, 10800, 11163, 12675, 13872, 14700, 15987, 16428, 16875, 18252, 19200, 20172, 21675, 22707, 23763, 24300, 24843, 27075, 28227, 30000, 30603

Offset: 1

Antonio Roldán, Sep 14 2017

nonn

			75 = 1^2 + 5^2 + 7^2 = 1 + 25 + 49, with 25 - 1 = 49 - 25 = 24.
675 = 3^2 + 15^2 + 21^2 = 9 + 225 + 441, with 225 - 9 = 441 - 225 = 216.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subsequence of A033428.

Cf. A000290, A000378, A004431, A134422, A292309, A292310, A292314, A292316.

t=4; k=3; while(t<=13000, i=k; e=0; v=t+i; while(i>1&&e==0, if(issquare(v), m=3*t; e=1; print1(m,", ")); i+=-2; v+=i); k+=2; t+=k)

Sequence is 3*(distinct elements in A198385).

Numbers of the form 3*m^2 where 2*m^2 is in A004431. - Chai Wah Wu, Oct 05 2017

A294594 Numbers that are the sum of three squares (square 0 allowed) in exactly five ways.

Original entry on oeis.org

146, 153, 185, 206, 221, 225, 230, 234, 257, 261, 266, 293, 305, 325, 338, 350, 353, 354, 362, 377, 381, 398, 402, 405, 409, 410, 413, 414, 419, 437, 470, 474, 477, 481, 491, 514, 525, 539, 557, 563, 579, 582, 584, 586, 590, 611, 612, 625, 630, 635, 638, 642

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 5.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly five ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..843

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 5 &]

Updated Mathematica program to Version 11. by Robert Price, Nov 01 2019

A351723 Numbers of the form x^2 + y^2 + z^2 + xyz with x,y,z nonnegative integers.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 14, 16, 17, 18, 20, 22, 25, 26, 28, 29, 32, 34, 36, 37, 38, 40, 41, 44, 45, 49, 50, 52, 53, 54, 58, 61, 62, 64, 65, 68, 70, 72, 73, 74, 76, 77, 80, 81, 82, 85, 88, 89, 90, 92, 94, 97, 98, 100, 101, 104, 106, 108, 109, 110, 112, 113, 116, 117, 118, 121, 122, 125, 128, 130, 133, 134, 136

Offset: 1

Zhi-Wei Sun, Feb 17 2022

nonn

It is easy to see that no term is congruent to 3 modulo 4.
Conjecture 1: a(n) < 2*n for all n > 0, and a(n)/n has a limit as n tends to the infinity. Also, a(n) <= a(n-1) + a(n-2) for all n > 4.
Conjecture 2: Let S = {x^2 + y^2 + z^2 + x*y*z: x,y,z = 0,1,2,...}.
(i) 7 and 487 are the only nonnegative integers which cannot be written as w^2 + s, where w is a nonnegative integer and s is an element of S. Also, 7, 87 and 267 are the only nonnegative integers which cannot be written as w^3 + s, where w is a nonnegative integer and s is an element of S.
(ii) Let k be 2 or 3. Then each nonnegative integer not congruent to 3 modulo 4 can be written as 4*w^k + s, where w is a nonnegative integer and s is an element of S.
This has been verified for nonnegative integers up to 10^6.

			a(3) = 2 with 2 = 1^2 + 1^2 + 0^2 + 1*1*0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Can each natural number be represented by 2*w^2+x^2+y^2+z^2+xyz with x,y,z in {0,1,2,...}? Question 416344 at MathOverflow, Feb. 17, 2022.

Cf. A000290, A000378, A000578, A351902.

tab={};Do[n=x^2+y^2+z^2+x*y*z;If[n<=140,tab=Append[tab,n]],{x,0,20},{y,0,x},{z,0,y}];Print[Sort[DeleteDuplicates[tab]]]

A351902 Number of ways to write n as w^2 + x^2 + y^2 + z^2 + 3xy*z, where w is a positive integer, and x,y,z are nonnegative integers with x <= y <= z.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 3, 3, 1, 2, 4, 3, 2, 2, 5, 4, 0, 3, 4, 5, 3, 1, 6, 4, 2, 1, 5, 5, 3, 5, 5, 5, 1, 3, 8, 4, 3, 2, 7, 7, 1, 3, 5, 7, 5, 3, 5, 9, 3, 4, 8, 3, 5, 1, 9, 8, 1, 2, 8, 9, 3, 5, 9, 6, 2, 5, 6, 8, 4, 6, 7, 7, 1, 3, 15, 6, 5, 5, 9, 9, 2, 4, 12, 9, 5, 2, 5, 10, 1, 5, 9, 8, 7, 5

Offset: 1

Zhi-Wei Sun, Feb 25 2022

nonn

Conjecture: We have a(n) > 0 except for n = 23. In other words, any positive integer n not equal to 23 can be written as w^2 + x^2 + y^2 + z^2 + 3*x*y*z, where w is a positive integer and x,y,z are nonnegative integers.

It seems that a(n) > 1 for all n > 695.

			a(60)  = 1 with  60 =  2^2 + 1^2 + 1^2 +  6^2 + 3*1*1*6.
a(128) = 1 with 128 =  8^2 + 0^2 + 0^2 +  8^2 + 3*0*0*8.
a(303) = 1 with 303 = 11^2 + 1^2 + 1^2 + 12^2 + 3*1*1*12.
a(359) = 1 with 359 =  3^2 + 1^2 + 5^2 + 12^2 + 3*1*5*12.
a(383) = 1 with 383 = 11^2 + 1^2 + 3^2 + 12^2 + 3*1*3*12.
a(695) = 1 with 695 = 17^2 + 1^2 + 9^2 +  9^2 + 3*1*9*9.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000118, A000290, A000378, A351723.

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

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A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

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A047449 Numbers that are primitively represented by x^2 + y^2 + z^2.

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A034043 Numbers that are imprimitively represented by x^2+y^2+z^2.

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A034045 Numbers that are imprimitively but not primitively represented by x^2+y^2+z^2.

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A034047 Numbers that are primitively or imprimitively represented by x^2+y^2+z^2, but not both.

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A294577 Numbers that are the sum of three squares (square 0 allowed) in exactly four ways.

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A292313 Numbers that are the sum of three squares in arithmetic progression.

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A294594 Numbers that are the sum of three squares (square 0 allowed) in exactly five ways.

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A351723 Numbers of the form x^2 + y^2 + z^2 + x*y*z with x,y,z nonnegative integers.

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A351723 Numbers of the form x^2 + y^2 + z^2 + xyz with x,y,z nonnegative integers.

A351902 Number of ways to write n as w^2 + x^2 + y^2 + z^2 + 3xy*z, where w is a positive integer, and x,y,z are nonnegative integers with x <= y <= z.