A000378 -id:A000378 - cp's OEIS frontend

A294710 Numbers that are the sum of three squares (square 0 allowed) in exactly seven ways.

306, 314, 341, 441, 450, 458, 494, 506, 581, 585, 593, 605, 654, 657, 674, 698, 706, 726, 731, 738, 746, 773, 806, 842, 850, 873, 890, 891, 893, 894, 899, 901, 905, 906, 934, 978, 985, 998, 1011, 1013, 1019, 1050, 1058, 1061, 1067, 1073, 1086, 1094, 1101

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 7.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly seven 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..988

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

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 7 &]

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

A133102 Number of partitions of n^3 into n distinct nonzero squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 5, 20, 56, 112, 268, 618, 1922, 8531, 29021, 100407, 321531, 899618, 2937312, 9295401, 31615059, 117365818, 403433963, 1417579281, 4848439367, 15960316056, 55180971700, 190251417034, 670818005444, 2429973932322

Offset: 1

Hugo Pfoertner, Sep 12 2007

nonn

			a(6) = 3 because there are 3 ways to express 6^3 = 216 as a sum of 6 distinct nonzero squares: 216 = 1^2 + 2^2 + 4^2 + 5^2 + 7^2 + 11^2 = 1^2 + 3^2 + 5^2 + 6^2 + 8^2 + 9^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 9^2.

Robert Gerbicz, May 09 2008, Table of n, a(n) for n = 1..40

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

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

a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

More terms from Robert Gerbicz, May 09 2008

A133103 Number of partitions of n^3 into n nonzero squares.

Original entry on oeis.org

1, 1, 2, 1, 10, 34, 156, 734, 3599, 18956, 99893, 548373, 3078558, 17510598, 101960454, 599522778, 3565904170, 21438347021, 129905092421, 794292345434, 4890875249113, 30326545789640, 189195772457341, 1187032920371427

Offset: 1

Hugo Pfoertner, Sep 11 2007

nonn

			a(2)=1 because the only way to express 2^3 = 8 as a sum of two squares is 8 = 2^2 + 2^2.
a(3)=2 because 3^3 = 27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2.

Robert Gerbicz, May 09 2008, Table of n, a(n) for n = 1..40

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

Cf. A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares), A133104 (number of ways to express n^4 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^3; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

More terms from Robert Gerbicz, May 09 2008

A133105 Number of partitions of n^4 into n distinct nonzero squares.

Original entry on oeis.org

1, 0, 1, 0, 21, 266, 2843, 55932, 884756, 13816633, 283194588, 5375499165, 125889124371, 3202887665805, 80542392920980, 2270543992935431, 64253268814048352, 1892633465941308859, 59116753827795287519, 1886846993941912938452

Offset: 1

Hugo Pfoertner, Sep 12 2007

nonn

			a(3)=1 because there is exactly one way to express 3^4 as the sum of 3 distinct nonzero squares: 81 = 1^2 + 4^2 + 8^2.

Robert Gerbicz, May 09 2008, Table of n, a(n) for n = 1..20

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

Cf. A133104 (number of ways to express n^4 as a sum of n nonzero squares), A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares).

a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

a(10) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

a(11) onwards from Robert Gerbicz, May 09 2008

A237708 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, partially encircled along the edge of the first octant of a sphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

0, 1, 1, 4, 4, 7, 6, 7, 7, 10, 10, 16, 13, 16, 16, 19, 16, 22, 22, 28, 25, 28, 27, 28, 28, 34, 28, 34, 34, 37, 34, 43, 40, 46, 43, 46, 46, 52, 46, 52, 49, 52, 49, 52, 52, 61, 55, 67, 63

Offset: 0

Rajan Murthy, Feb 11 2014

nonn

			At radius 0, there are no partially filled cubes.  At radius >0 but < sqrt(1), there is 1 partially filled square along the edge of the sphere.  At radius = sqrt(1), there is 1 partially filled cube along the edge of the sphere.  At radius > 1 but < sqrt(2), there  are 4 partially filled cubes.

Cf. A000378 (corresponds to the square radius of alternate entries).

Cf. A234300 (2-dimensional analog).

A294711 Numbers that are the sum of three squares (square 0 allowed) in exactly eight ways.

Original entry on oeis.org

369, 374, 446, 461, 486, 509, 530, 549, 566, 621, 641, 666, 677, 686, 710, 749, 770, 789, 797, 818, 821, 825, 833, 849, 869, 882, 902, 945, 954, 962, 969, 971, 981, 1010, 1014, 1017, 1022, 1029, 1069, 1085, 1098, 1146, 1157, 1174, 1221, 1233, 1242, 1245

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 8.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly eight 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..1701

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

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 8 &]

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

A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

14, 29, 35, 50, 56, 66, 77, 83, 93, 107, 110, 116, 126, 140, 149, 155, 158, 165, 179, 194, 197, 200, 210, 219, 224, 242, 245, 251, 261, 264, 275, 290, 293, 302, 308, 315, 318, 332, 341, 350, 365, 371, 372, 381, 395, 398, 413, 428, 434, 435, 440, 450, 461, 462, 464, 482

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

			35 = 1^2 + 3^2 + 5^2, with 3 - 1 = 5 - 3 = 2;
371 = 1^2 + 9^2 + 17^2, with 9 - 1 = 17 - 9 = 8. Also 371 = 9^2 + 11^2 + 13^2, with 11 - 9 = 13 - 11 = 2.

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

Cf. A000290, A000378, A000408, A085317, A120328, A292313, A306213, A306214.

N:= 1000: # for terms <= N
S:= {seq(seq(3*a^2+2*b^2, b=1..min(a-1, floor(sqrt((N-3*a^2)/2)))),a=1..floor(sqrt(N/3)))}:
sort(convert(S,list)); # Robert Israel, Jun 08 2020

for(n=3, 600, k=sqrt(n/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^2)/2; if(b==truncate(b)&&issquare(b), d=sqrt(b); if(d>=1&&d<=a-1, v=1; print1(n,", "))); a+=1))

w=List(); for(n=3, 600, k=sqrt(n/3); for(a=2, k, for(c=1, a-1, v=(a-c)^2+a^2+(a+c)^2; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))

A308661 Number of ways to write 12n+5 as (2^a5^b)^2 + c^2 + d^2, where a,b,c,d are nonnegative integers with a > 0 and c <= d.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Jun 15 2019

nonn

Conjecture 1: a(n) > 0 for all n = 0,1,2,.... Equivalently, for each nonnegative integer n we can write 3*n+1 as a*(a+1)/2 + b*(b+1)/2 + (2^c*5^d)^2 with a,b,c,d nonnegative integers.
Conjecture 2: For each n = 0,1,2,... we can write 24*n+10 as a^2 + b^2 + (2^c*3^d)^2 with a,b,c,d nonnegative integers and d > 0.
We have verified Conjectures 1 and 2 for n up to 2*10^8 and 10^8 respectively.
By the Gauss-Legendre theorem on sums of three squares, for each n = 0,1,... we can write 4*n+1 (or 4*n+2, or 8*n+3) as the sum of three squares.
Conjecture 1 holds for n < 8.33*10^9. - Giovanni Resta, Jun 19 2019

			a(0) = 1 with 12*0 + 5 = (2^1*5^0)^2 + 0^2 + 1^2.
a(4) = 2 with 12*4 + 5 = 53 = (2^1*5^0)^2 + 0^2 + 7^2 = (2^2*5^0)^2 + 1^2 + 6^2.
a(441019) = 2 with 12*441019 + 5 = 5292233 = (2^1*5^2)^2 + 513^2 + 2242^2 = (2^3*5^1)^2 + 757^2 + 2172^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000079, A000217, A000244, A000290, A000351, A000378, A001481, A308566, A308584, A308621, A308623, A308640, A308641, A308644, A308656, A308662.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[12n+5-4^a*25^b-x^2],r=r+1],{a,1,Log[4,12n+5]},{b,0,Log[25,(12n+5)/4^a]},{x,0,Sqrt[(12n+5-4^a*25^b)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

Original entry on oeis.org

1, 3, 3, 1, 2, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3

Offset: 1

Yifan Xie, Apr 05 2023

a(n) is the smallest number k such that n*k can be expressed as the sum of k nonzero squares.

			For n = 2, if k = 1, 2*1 = 2 is a nonsquare; if k = 2, 2*2 = 4 cannot be expressed as the sum of 2 nonzero squares; if k = 3, 2*3 = 6 = 2^2+1^2+1^2, so a(2) = 3.

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

Yifan Xie, Table of n, a(n) for n = 1..10000
Wikipedia, Lagrange's four-square theorem
Index entries for sequences related to sums of squares

Cf. A000290, A000378, A000404, A000408, A000414, A000534, A047700.

Cf. A362068 (allows zeros), A362110 (distinct).

findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k-1, m-j*j), return(1)))));
a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));

a(n) <= 4. Proof: With Lagrange's four-square theorem, if 4*n is not the sum of 4 positive squares (see A000534), then it is easy to express 3*n as the sum of 3 positive squares. - Yifan Xie and Thomas Scheuerle, Apr 29 2023

A047809 a(n) counts different values of i^2+j^2+k^2 <= n^2 or number of distances from the origin to all integer points inside a sphere of radius n.

Original entry on oeis.org

1, 2, 5, 9, 15, 23, 32, 43, 55, 70, 86, 103, 122, 143, 166, 190, 215, 243, 273, 304, 336, 371, 406, 443, 482, 523, 566, 611, 656, 704, 753, 803, 855, 910, 966, 1024, 1083, 1145, 1207, 1270, 1336, 1404, 1474, 1544, 1616, 1690, 1766, 1843, 1922, 2004

Offset: 0

Wouter Meeussen,_David W. Wilson_

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A047808, A000378.

Table[ Length@Union@Flatten@Table[ i^2+j^2+k^2, {i, 0, n}, {j, 0, Min[ i, Floor[ Sqrt[ n^2-i^2 ] ] ]}, {k, 0, Min[ j, Floor[ Sqrt[ n^2-i^2-j^2 ] ] ]} ], {n, 0, 64} ]

a(n) = number of b(i) <= n^2, b() = A000378.

cp's OEIS Frontend

A294710 Numbers that are the sum of three squares (square 0 allowed) in exactly seven ways.

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A133102 Number of partitions of n^3 into n distinct nonzero squares.

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PARI

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A133103 Number of partitions of n^3 into n nonzero squares.

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PARI

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A133105 Number of partitions of n^4 into n distinct nonzero squares.

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A237708 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, partially encircled along the edge of the first octant of a sphere centered at the origin, ordered by increasing radius.

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A294711 Numbers that are the sum of three squares (square 0 allowed) in exactly eight ways.

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Mathematica

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A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

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Maple

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PARI

A308661 Number of ways to write 12*n+5 as (2^a*5^b)^2 + c^2 + d^2, where a,b,c,d are nonnegative integers with a > 0 and c <= d.

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Mathematica

A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

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A308661 Number of ways to write 12n+5 as (2^a5^b)^2 + c^2 + d^2, where a,b,c,d are nonnegative integers with a > 0 and c <= d.