A000378 -id:A000378 - cp's OEIS frontend

A074590 Number of primitive solutions to n = x^2 + y^2 + z^2 (i.e., with gcd(x,y,z) = 1).

1, 6, 12, 8, 0, 24, 24, 0, 0, 24, 24, 24, 0, 24, 48, 0, 0, 48, 24, 24, 0, 48, 24, 0, 0, 24, 72, 24, 0, 72, 48, 0, 0, 48, 48, 48, 0, 24, 72, 0, 0, 96, 48, 24, 0, 48, 48, 0, 0, 48, 72, 48, 0, 72, 72, 0, 0, 48, 24, 72, 0, 72, 96, 0, 0, 96, 96, 24, 0, 96, 48, 0, 0, 48, 120

Offset: 0

N. J. A. Sloane, Dec 03 2002

			G.f. = 1 + 6*x + 12*x^2 + 8*x^3 + 24*x^5 + 24*x^6 + 24*x^9 + 24*x^10 + 24*x^11 + ...

See A005875 for references.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.

Cf. A005875 (all solutions).

a[n_] := (r = Reduce[ GCD[x, y, z] == 1 && n == x^2 + y^2 + z^2, {x, y, z}, Integers]; If[ r === False, 0, Length[ {ToRules[r]} ] ] ); a[0] = 1; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 13 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ Select[ {x, y, z} /. FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9], 1 == GCD @@ # &]]; (* Michael Somos, May 21 2015 *)

n is representable as the sum of 3 squares if and only if n is not of the form 4^a (8k + 7) (cf. A000378).
A005875(n) = Sum_{d^2|n} a(n/d^2).
Let h = number of classes of primitive binary quadratic forms, corresponding to the discriminant D = -n if n = 3 (mod 8), D = -4n if n = 1, 2, 5, 6 (mod 8) and let d_1 = 1/2, d_3 = 1/3, d_n = 1 otherwise. Then a(n) = 12 h d_n, if n = 1, 2, 5, 6 (mod 8), 24 h d_n, if n = 3 (mod 8). (Grosswald)
Also, if n is squarefree and (r/n) is the Jacobi symbol, a(n) = 24 sum(r = 1, [n/4], (r/n)) if n = 1 (mod 4), 8 sum(r = 1, [n/2], (r/n)) if n = 3 (mod 8). (Grosswald)

More terms from Vladeta Jovovic, Dec 04 2002

A133104 Number of partitions of n^4 into n nonzero squares.

Original entry on oeis.org

1, 0, 3, 1, 49, 732, 9659, 190169, 3225654, 61896383, 1360483727, 30969769918, 778612992660, 20749789703573, 579672756740101, 17115189938667708, 525530773660159970, 16825686497823918869, 561044904645283065043, 19368002907483932784642

Offset: 1

Hugo Pfoertner, Sep 11 2007

nonn

			a(3)=3 because there are 3 ways to express 3^4 = 81 as a sum of 3 nonzero squares: 81 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2.
a(4)=1 because the only way to express 4^4 = 256 as a sum of 4 nonzero squares is 256 = 8^2 + 8^2 + 8^2 + 8^2.

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.

Cf. A133105 (number of ways to express n^4 as a sum of n distinct nonzero squares), A133103 (number of ways to express n^3 as a sum of n nonzero squares).

a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^4; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

a(9) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

a(10) onwards from Robert Gerbicz, May 09 2008

A255844 a(n) = 2*n^2 + 6.

Original entry on oeis.org

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
Equivalently, numbers m such that 2*m-12 is a square.
For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
Cf. A152811: nonnegative numbers of the form 2*m^2-6.
Subsequence of A000378.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+6: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 6, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+6)
```
Sage
```
[2*n^2+6 for n in (0..50)]
```

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117950(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Corrected and extended by Bruno Berselli, Mar 11 2015

A294712 Numbers that are the sum of three squares (square 0 allowed) in exactly nine ways.

Original entry on oeis.org

425, 521, 545, 569, 614, 650, 701, 725, 729, 774, 809, 810, 845, 857, 953, 974, 989, 990, 1053, 1062, 1070, 1074, 1091, 1118, 1134, 1139, 1166, 1179, 1217, 1249, 1251, 1262, 1266, 1277, 1298, 1310, 1418, 1446, 1458, 1470, 1525, 1541, 1546, 1571, 1594, 1611

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 9.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly nine 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.

			545 =  8^2 + 15^2 + 16^2
    =  0^2 + 16^2 + 17^2
    = 10^2 + 11^2 + 18^2
    =  5^2 + 14^2 + 18^2
    =  8^2 +  9^2 + 20^2
    =  1^2 + 12^2 + 20^2
    =  2^2 + 10^2 + 21^2
    =  5^2 +  6^2 + 22^2
    =  0^2 +  4^2 + 23^2. - _Robert Israel_, Nov 08 2017

Robert Price, Table of n, a(n) for n = 1..1105

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594, A294595, A294710, A294711.

N:= 10000: # to get all terms <= N
V:= Array(0..N):
for i from 0 to isqrt(N) do
  for j from 0 to i while i^2 + j^2 <= N do
    for k from 0 to j while i^2 + j^2 + k^2 <= N do
      t:= i^2 + j^2 + k^2;
      V[t]:= V[t]+1;
od od od:
select(t -> V[t] = 9, [$1..N]); # Robert Israel, Nov 08 2017

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 9 &]

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

A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive squares.

Original entry on oeis.org

25, 100, 400, 1600, 6400, 25600, 102400, 409600, 1638400, 6553600, 26214400, 104857600, 419430400, 1677721600, 6710886400, 26843545600, 107374182400, 429496729600, 1717986918400, 6871947673600, 27487790694400, 109951162777600, 439804651110400, 1759218604441600

Offset: 1

Bernard Schott, Aug 17 2019

This sequence comes from the study of A309778, exactly, A309778(n) = 2 iff n^2 belongs to this sequence here.
According to Draxl link, a(n) is a term of this sequence iff a(n) = 5^2 * 4^(n-1) with n >= 1.
This sequence is a subsequence of A219222 whose terms are all of the form b_0 * 4^k with b_0 in A051952, hence, the only primitive term of this sequence here is 25.

			25 = 5^2 = 3^2 + 4^2,
100 = 10^2 = 6^2 + 8^2,
5^2 * 4^(n-1) = (5 * 2^(n-1))^2 = (3 * 2^(n-1))^2 + (4 * 2^(n-1))^2, but these terms are not the sum of three positive squares.

H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés., Mémoires de la SMF, tome 37 (1974), p. 53-53.
Index entries for linear recurrences with constant coefficients, signature (4).

Intersection of A000290 and A219222.

Cf. A000302, A000378, A000408, A051952, A134422, A309778.

Array[25*4^(# - 1) &, 24] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 25 * 4^(n-1); \\ Jinyuan Wang, Aug 18 2019

a(n) = 5^2 * 4^(n-1) with n >= 1.
a(n) = 4*a(n-1) for n > 1. G.f.: 25*x/(1 - 4*x). - Chai Wah Wu, Aug 29 2019
a(n) = 25 * A000302(n-1). - Alois P. Heinz, Aug 29 2019
E.g.f.: 25*(exp(4*x) - 1)/4. - Stefano Spezia, Oct 28 2023

A166265 Numbers of the form 1+x^2+y^2, x, y integers >= 1.

Original entry on oeis.org

3, 6, 9, 11, 14, 18, 19, 21, 26, 27, 30, 33, 35, 38, 41, 42, 46, 51, 53, 54, 59, 62, 66, 69, 73, 74, 75, 81, 83, 86, 90, 91, 98, 99, 101, 102, 105, 107, 110, 114, 117, 118, 123, 126, 129, 131, 137, 138, 146, 147, 149, 150, 154, 158, 161, 163, 165, 170, 171, 174, 179, 181, 182

Offset: 1

N. J. A. Sloane, Mar 05 2010

nonn

Cf. A000408, A000378, A005767, A169580, A166687.

With[{nn=40},Take[Union[Total/@Tuples[Range[nn]^2,2]+1],2*nn]] (* Harvey P. Dale, Mar 12 2015 *)

A173256 Partial sums of A001481.

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 29, 39, 52, 68, 85, 103, 123, 148, 174, 203, 235, 269, 305, 342, 382, 423, 468, 517, 567, 619, 672, 730, 791, 855, 920, 988, 1060, 1133, 1207, 1287, 1368, 1450, 1535, 1624, 1714, 1811, 1909, 2009, 2110, 2214, 2320, 2429, 2542, 2658, 2775

Offset: 1

Jonathan Vos Post, Feb 14 2010

nonn

The subsequence of primes in this sequence begins 3, 7, 29, 103, 269, 619, 1811, 3271.

			a(66) = 0 + 1 + 2 + 4 + 5 + 8 + 9 + 10 + 13 + 16 + 17 + 18 + 20 + 25 + 26 + 29 + 32 + 34 + 36 + 37 + 40 + 41 + 45 + 49 + 50 + 52 + 53 + 58 + 61 + 64 + 65 + 68 + 72 + 73 + 74 + 80 + 81 + 82 + 85 + 89 + 90 + 97 + 98 + 100 + 101 + 104 + 106 + 109 + 113 + 116 + 117 + 121 + 122 + 125 + 128 + 130 + 136 + 137 + 144 + 145 + 146 + 148 + 149 + 153 + 157 + 160 = 4876.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001481, A022544, A004018, A000161, A002654, A064533, A000404, A002828, A000378, A025284-A025320, A125110, A091072.

N:= 1000:
A001481:= sort(convert({seq(seq(x^2+y^2, y=0..floor(sqrt(N-x^2))),x=0..floor(sqrt(N)))},list)):
ListTools:-PartialSums(A001481); # Robert Israel, Mar 15 2016

from itertools import count, accumulate, islice
from sympy import factorint
def A173256_gen(): # generator of terms
    return accumulate(filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0)))
A173256_list = list(islice(A173256_gen(),30)) # Chai Wah Wu, Jun 27 2022

a(n) = Sum_{i=1..n} A001481(i) = Sum_{i=1..n} (numbers that are the sum of 2 nonnegative squares) = Sum_{i=1..n} (numbers n such that i = x^2 + y^2 has a solution in nonnegative integers x, y).

a(21) corrected by Robert Israel, Mar 15 2016

A294713 Numbers that are the sum of three squares (square 0 allowed) in exactly ten ways.

Original entry on oeis.org

594, 626, 629, 734, 846, 914, 926, 929, 1001, 1026, 1041, 1097, 1125, 1190, 1193, 1209, 1214, 1229, 1241, 1265, 1289, 1326, 1329, 1382, 1386, 1409, 1433, 1490, 1505, 1509, 1521, 1530, 1581, 1637, 1689, 1691, 1713, 1725, 1730, 1739, 1749, 1754, 1770, 1778

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 10.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly ten 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..1835

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594, A294595, A294710, A294711, A294712.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 10 &]

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

A302359 Numbers that are the sum of 3 squares > 1.

Original entry on oeis.org

12, 17, 22, 24, 27, 29, 33, 34, 36, 38, 41, 43, 44, 45, 48, 49, 50, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 93, 94, 96, 97, 98, 99, 101, 102, 104, 105, 106, 107, 108, 109, 110, 113, 114, 115, 116, 117, 118, 120, 121, 122, 123, 125, 126, 129

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			33 is in the sequence because 33 = 2^2 + 2^2 + 5^2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000378, A000408, A085989, A214514, A281155.

max = 130; f[x_] := Sum[x^(k^2), {k, 2, 20}]^3; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, max}]]
With[{nn=15},Select[Union[Total/@Tuples[Range[2,nn]^2,3]],#<=nn^2+8&]] (* Harvey P. Dale, Jul 05 2021 *)

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(N):
    sqrs = list(takewhile(lambda x: x<=N, (i**2 for i in count(2))))
    sum3 = set(sum(c) for c in mc(sqrs, 3) if sum(c) <= N)
    return sorted(sum3)
print(aupto(129)) # Michael S. Branicky, Dec 17 2021

A347360 Numbers that can be represented as the sum of squares of 3 numbers and also equal to twice the sum of their joint products.

Original entry on oeis.org

18, 72, 98, 162, 288, 338, 392, 450, 648, 722, 882, 1152, 1352, 1458, 1568, 1800, 1922, 2178, 2450, 2592, 2738, 2888, 3042, 3528, 3698, 4050, 4608, 4802, 5202, 5408, 5832, 6272, 6498, 7200, 7442, 7688, 7938, 8450, 8712, 8978, 9522, 9800, 10368, 10658, 10952, 11250, 11552, 11858

Offset: 1

Alexander Kritov, Sep 22 2021

nonn

Integers that can be represented as the sum of three squares of integers x, y, z, and additionally also satisfy x^2+y^2+z^2 = k *(x*y+ x*z + y*z), with k=2.

All possible k are given by A331605.

			For example, the third term (1,4,9) is 1^2+4^2+9^2 = 2*(1*4+1*9+4*9) = 98.
The sequence is given by
   a(n)    (x, y, z)
    18     (1,1,4)
    72     (2,2,8)
    98     (1,4,9)
   162     (3,3,12)
   288     (4,4,16)
   338     (1,9,16)
   392     (2,8,18)
   450     (5,5,20)
   648     (6,6,24)
   722     (4,9,25)
   882     (1,16,25) (3,12,27)  (7,7,28)
  1152     (8,8,32)  (2,18,32)
  1352     (2,18,32)
  1458     (9,9,36)
  1568     (4,16,36)
  1800     (10,10,40)
  1922     (1,25,36)
  2178     (11,11,44)
  2450     (5,20,45)
  2592     (12,12,48)
  2738     (9,16,49)
  2888     (8,18,50)
  3042     (3,27,48) (4,25,49) (13,13,52)
  3528     (2,32,50) (6,24,54)

E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.

Subsequence of A000378. Complement of A004215.

Cf. A033428 (case k=1), A324929, A331605 (k-numbers).

q[n_] := (s = Select[PowersRepresentations[n,3,2], AllTrue[#, #1 > 0 &]&]) != {} && MemberQ[(#[[1]]*#[[2]] + #[[2]]*#[[3]] + #[[3]]*#[[1]])& /@ s, n/2]; Select[Range[2, 12000, 2], q] (* Amiram Eldar, Oct 03 2021 *)

Empirically, such numbers appear to be a(n) = 2*b_n^2 where b_n are numbers whose product of prime indices is even (A324929).The triplet (x,y,x) is always (n*k^2, n*m^2, n*p^2).

cp's OEIS Frontend

A074590 Number of primitive solutions to n = x^2 + y^2 + z^2 (i.e., with gcd(x,y,z) = 1).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A133104 Number of partitions of n^4 into n nonzero squares.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A255844 a(n) = 2*n^2 + 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A294712 Numbers that are the sum of three squares (square 0 allowed) in exactly nine ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A166265 Numbers of the form 1+x^2+y^2, x, y integers >= 1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A173256 Partial sums of A001481.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

Extensions