A000378 -id:A000378 - cp's OEIS frontend

A034044 Numbers that are primitively but not imprimitively represented by x^2+y^2+z^2.

1, 2, 3, 5, 6, 10, 11, 13, 14, 17, 19, 21, 22, 26, 29, 30, 33, 34, 35, 37, 38, 41, 42, 43, 46, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 73, 74, 77, 78, 82, 83, 85, 86, 89, 91, 93, 94, 97, 101, 102, 105, 106, 107, 109, 110, 113, 114, 115, 118, 122, 123

Offset: 1

N. J. A. Sloane

nonn

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

Union of A034043 and A034044 is A000378. Union of A034044 and A034045 is A034047. Numbers that are in A000378 and squarefree. - Ray Chandler, Sep 05 2004

A034046 Numbers that are both primitively and imprimitively represented by x^2+y^2+z^2.

Original entry on oeis.org

9, 18, 25, 27, 45, 49, 50, 54, 75, 81, 90, 98, 99, 117, 121, 125, 126, 147, 150, 153, 162, 169, 171, 189, 198, 225, 234, 242, 243, 245, 250, 261, 270, 275, 289, 294, 297, 306, 315, 325, 333, 338, 342, 350, 361, 363, 369, 378, 387, 405, 414, 425

Offset: 1

N. J. A. Sloane

nonn

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

Union of A034046 and A034047 is A000378. Intersection of A047449 and A034043 is A034046. Numbers that are in A000378 and neither squarefree nor congruent to 0 mod 4. - Ray Chandler, Sep 05 2004

A072400 (Factors of 4 removed from n) modulo 8.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 1, 2, 3, 3, 5, 6, 7, 1, 1, 2, 3, 5, 5, 6, 7, 6, 1, 2, 3, 7, 5, 6, 7, 2, 1, 2, 3, 1, 5, 6, 7, 2, 1, 2, 3, 3, 5, 6, 7, 3, 1, 2, 3, 5, 5, 6, 7, 6, 1, 2, 3, 7, 5, 6, 7, 1, 1, 2, 3, 1, 5, 6, 7, 2, 1, 2, 3, 3, 5, 6, 7, 5, 1, 2, 3, 5, 5, 6, 7, 6, 1, 2, 3, 7, 5, 6, 7, 6

Offset: 1

Reinhard Zumkeller, Jun 16 2002

a(n) <> 7 iff n equals the sum of 3 integer squares.

a(A004215(k)) = 7 for k>0;

			From _Michael De Vlieger_, May 08 2017: (Start)
a(4) = 1 since 4 = 1 * 4^1 and 4 / 4^1 = 1; 1 = 1 (mod 8).
a(5) = 5 since it is not a multiple of 4; 5 = 5 (mod 8).
a(12) = 3 since 12 = 3 * 4^1 and 12 / 4^1 = 3; 3 = 3 (mod 8).
a(44) = 3 since 44 = 11 * 4^1 and 44 / 4^1 = 11; 3 = 11 (mod 8).
a(64) = 1 since 64 = 1 * 4^3 and 64 / 4^3 = 1; 1 = 1 (mod 8). (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Numbers.

Cf. A000378, A004215, A057427, A065883, A072401, A286366.

Array[Mod[If[Mod[#, 4] == 0, #/4^IntegerExponent[#, 4], #], 8] &, 96] (* Michael De Vlieger, May 08 2017 *)

a(n) = (n >> (2*valuation(n, 4))) % 8; \\ Amiram Eldar, May 15 2025

def A072400(n): return (n>>((~n&n-1).bit_length()&-2))&7 # Chai Wah Wu, Aug 01 2023

a(n) = A065883(n) mod 8.
A072401(n) = 1 - A057427(7 - a(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, May 15 2025

Offset corrected (from 0 to 1) by Antti Karttunen, May 08 2017

A125084 Cubes which have a partition as the sum of 3 squares.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 512, 729, 1000, 1331, 1728, 2197, 2744, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 13824, 15625, 17576, 19683, 24389, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 64000, 68921, 74088, 79507, 85184, 91125

Offset: 1

Artur Jasinski, Nov 20 2006, Nov 21 2006, Nov 22 2006

nonn

If n is of the form 4^i*(8j+7) (where i>=0, j>=0) then n^3 is not in the sequence because n^3 is of the mentioned form so n^3 is in A004215 hence according to the definition n^3 is not in this sequence (see formula for A004215). Hence 7^3, 15^3, 23^3, 28^3, 31^3, 39^3, ... are not in the sequence. Is there a number n such that n^3 is not in the sequence but n is not of the form 4^i*(8j+7)? - Farideh Firoozbakht, Nov 23 2006
A number n^3 belongs to this sequence if and only if n is sum of three squares. Proof is immediate from Catalan's identity (x^2 + y^2 + z^2)^3 = x^2*(3*z^2 - x^2 - y^2)^2 + y^2*(3*z^2 - x^2 - y^2)^2 + z^2*(z^2 - 3*x^2 - 3*y^2)^2. - Artur Jasinski, Dec 09 2006
If n = a^2 + b^2 + c^2, then n^3 = (n*a)^2 + (n*b)^2 + (n*c)^2. Conversely, suppose there were an n such that n^3 is in A000378 but n is not. Then n must be of form 4^k*(8i+7). But n^3 would also be of the form 4^k*(8i+7) and thus n^3 would not be in A000378, contradicting the original assumption. This argument is easily extended to all odd powers, i.e., n^(2k+1) is in A000378 iff n is in A000378. - Ray Chandler, Feb 03 2009

			125 is in the sequence because
  125 = 5^3 = 0^2 + 2^2 + 11^2
            = 0^2 + 5^2 + 10^2
            = 3^2 + 4^2 + 10^2
            = 5^2 + 6^2 +  8^2.
   27 = 3^3 = 1^2 + 1^2 +  5^2, so  27 is a term.
  125 = 5^3 = 0^2 + 2^2 + 11^2, so 125 is a term.
  216 = 6^3 = 2^2 + 4^2 + 14^2, so 216 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A000378, A000578, A004215.

Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* Ray Chandler, Nov 23 2006 *)

isA125084(n)={ local(cnt,a,b) ; cnt=0 ; a=0; while(a^2<=n, b=0 ; while(b<=a && a^2+b^2<=n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ; } { for(n=1,300, if(isA125084(n^3), print1(n^3,", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 23 2006

a(n) = A000378(n)^3.

Equals A000578 INTERSECT A000378.

Corrected and extended by Farideh Firoozbakht, Ray Chandler and R. J. Mathar, Nov 23 2006

A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

Original entry on oeis.org

194, 209, 269, 281, 290, 297, 321, 326, 329, 342, 365, 386, 389, 401, 426, 434, 449, 459, 482, 485, 489, 497, 513, 531, 534, 542, 546, 554, 558, 561, 578, 601, 602, 633, 649, 659, 662, 665, 675, 678, 681, 693, 699, 705, 713, 714, 722, 737, 741, 747, 750, 754

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 6.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly six 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..1400

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

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 6 &]

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

A004014 Norms of vectors in the b.c.c. lattice.

Original entry on oeis.org

0, 3, 4, 8, 11, 12, 16, 19, 20, 24, 27, 32, 35, 36, 40, 43, 44, 48, 51, 52, 56, 59, 64, 67, 68, 72, 75, 76, 80, 83, 84, 88, 91, 96, 99, 100, 104, 107, 108, 115, 116, 120, 123, 128, 131, 132, 136, 139, 140, 144, 147, 148, 152, 155, 160, 163, 164, 168

Offset: 0

N. J. A. Sloane

Integers such that A004013(n) is nonzero. - Michael Somos, Jul 28 2014
A subsequence of A047458. The complement seems to be 4*A004215. - Andrey Zabolotskiy, Nov 11 2021
From Mohammed Yaseen, Nov 06 2022: (Start)
These are numbers of the form x^2+y^2+z^2 where x, y and z are either all even (including zero) or all odd.
The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in an f.c.c. lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A000378 for simple cubic lattice. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116. (Chapter 4 section 6.7)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to b.c.c. lattice

Cf. A004013, A047458, A004215.

Union of A034045 and A017101. - Mohammed Yaseen, Nov 06 2022

f:= JacobiTheta2(0,z^4)^3+JacobiTheta3(0,z^4)^3:
S:= series(f,z,1001):
select(t -> coeff(S,z,t) <> 0, [$0..1000]); # Robert Israel, Oct 18 2015

f = EllipticTheta[2, 0, z^4]^3 + EllipticTheta[3, 0, z^4]^3; S = f + O[z]^200; Flatten[Position[CoefficientList[S, z], ?Positive] - 1] (* _Jean-François Alcover, Oct 23 2016, after Robert Israel *)

More terms from Sean A. Irvine, Oct 17 2015

A124978 Smallest positive number which has exactly n different partitions as a sum of 4 squares x^2+y^2+z^2+t^2.

Original entry on oeis.org

1, 4, 18, 34, 50, 66, 82, 114, 90, 130, 150, 178, 162, 198, 318, 210, 250, 234, 322, 406, 465, 330, 306, 402, 462, 390, 474, 378, 490, 486, 654, 610, 522, 450, 778, 678, 642, 570, 666, 726, 594, 714, 770, 774, 986, 630, 738, 945, 1035, 850, 1222, 978, 1014, 918

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

Is it known that a(n) always exists? - Franklin T. Adams-Watters, Dec 18 2006

A002635(a(n)) = n. - Reinhard Zumkeller, Jul 13 2014

			a(4)=34 because 34 is smallest number which has 4 partitions 34=4^2+3^2+3^2+0^2 = 4^2+4^2+1^2+1^2 = 5^2+2^2+2^2+1^2 = 5^2+3^2+0^2+0^2
a(3)=18 which has 3 partitions 18=0^2+0^2+3^2+3^2=0^2+1^2+1^2+4^2=1^2+2^2+2^2+3^2.

Cf. A006431, A094942, A124979-A124983, A000378, A002635, A061262.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a124978 = (+ 1) . fromJust . (`elemIndex` (tail a002635_list))
-- Reinhard Zumkeller, Jul 13 2014

kmin[n_] := If[n<5, 1, 10n](* empirical, should be lowered in case of doubt *);
a[n_] := a[n] = For[k=kmin[n], True, k++, If[Length[PowersRepresentations[ k, 4, 2]] == n, Return[k]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 1000}] (* Jean-François Alcover, Mar 11 2019 *)

cnt4sqr(n)={ local(cnt=0,t2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), for(z=y,floor(n-x^2-y^2), t2=n-x^2-y^2-z^2 ; if( t2>=z^2 && issquare(n-x^2-y^2-z^2), cnt++ ; ) ; ) ; ) ; ) ; return(cnt) ; }
A124978(n)= { local(a=1) ; while(1, if( cnt4sqr(a)==n, return(a) ; ) ; a++ ; ) ; }
{ for(n=1,100, print(n," ",A124978(n)) ; ) ; } \\ R. J. Mathar, Nov 29 2006

Corrected and extended by R. J. Mathar, Nov 29 2006

More terms from Franklin T. Adams-Watters, Dec 18 2006

A166687 Numbers of the form x^2 + y^2 + 1, x, y integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 14, 17, 18, 19, 21, 26, 27, 30, 33, 35, 37, 38, 41, 42, 46, 50, 51, 53, 54, 59, 62, 65, 66, 69, 73, 74, 75, 81, 82, 83, 86, 90, 91, 98, 99, 101, 102, 105, 107, 110, 114, 117, 118, 122, 123, 126, 129, 131, 137, 138, 145, 146, 147, 149, 150, 154, 158, 161

Offset: 1

N. J. A. Sloane, Mar 05 2010

nonn

A001481 is the main entry for this sequence.

As Ng points out (Lemma 2.2), each prime divides some member of this sequence: 2 divides a(2) = 2, 3 divides a(3) = 3, 5 divides a(4) = 5, 7 divides a(9) = 14, etc. - Charles R Greathouse IV, Jan 04 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Jia Hong Ray Ng, Quarternions and the four square theorem, 2008 Summer VIGRE Program for Undergraduates
Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2 + y^2 + 1, Monatshefte für Mathematik vol. 189 (2019), pp. 715-733. arXiv:1708.08629 [math.NT]

Cf. A000408, A000378, A005767, A169580, A166265, A079545.

N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2+y^2+1,y=0..floor(sqrt(N-1-x^2))),x=0..floor(sqrt(N-1)))}:
sort(convert(S,list)); # Robert Israel, Jan 05 2016

Select[Range@ 162, Resolve[Exists[{x, y}, Reduce[# == x^2 + y^2 + 1, {x, y}, Integers]]] &] (* Michael De Vlieger, Jan 05 2016 *)

is(n)=my(f=factor(n-1)); for(i=1, #f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2016

list(lim)=my(v=List(),t); lim\=1; for(m=0,sqrtint(lim-1), t=m^2+1; for(n=0, min(sqrtint(lim-t),m), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Jan 05 2016

A261904 Largest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, -1, 2, 3, 3, 3, 2, 3, 3, -1, 4, 4, 4, 3, 4, 4, 3, -1, 4, 5, 5, 5, -1, 5, 5, -1, 4, 5, 5, 5, 6, 6, 6, -1, 6, 6, 5, 5, 6, 6, 6, -1, 4, 7, 7, 7, 6, 7, 7, -1, 6, 7, 7, 7, -1, 6, 7, -1, 8, 8, 8, 7, 8, 8, 6, -1, 8, 8, 8, 7, 6, 8, 7, -1, 8, 9, 9

Offset: 0

N. J. A. Sloane, Sep 08 2015

sign

a(n) = -1 iff n is in A004215, a(n) >= 0 iff n is in A000378.

Somehow maximizing x seems like the right thing to do (since it is natural to try a greedy algorithm first). If we minimize x we get A261915.

			Tabls showing initial values of n,x,y,z:
0 0 0 0
1 1 0 0
2 1 1 0
3 1 1 1
4 2 0 0
5 2 1 0
6 2 1 1
7 -1 -1 -1
8 2 2 0
9 3 0 0
10 3 1 0
11 3 1 1
12 2 2 2
13 3 2 0
14 3 2 1
15 -1 -1 -1
16 4 0 0
17 4 1 0
18 4 1 1
19 3 3 1
20 4 2 0
...

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program
Index entries for sequences related to sums of squares

Cf. A000378, A004215, A005875, A261915.

Analogs for 4 squares: A178786 and A122921.

More terms from David Consiglio, Jr., Sep 08 2015

A261915 Smallest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, -1, 2, 2, 3, 3, 2, 3, 3, -1, 4, 3, 3, 3, 4, 4, 3, -1, 4, 4, 4, 3, -1, 4, 5, -1, 4, 4, 4, 5, 4, 6, 5, -1, 6, 4, 5, 5, 6, 5, 6, -1, 4, 6, 5, 5, 6, 6, 5, -1, 6, 5, 7, 5, -1, 6, 6, -1, 8, 6, 5, 7, 6, 7, 6, -1, 6, 6, 7, 5, 6, 6, 7, -1, 8, 6, 8

Offset: 0

N. J. A. Sloane, Sep 11 2015

sign

a(n) = -1 iff n is in A004215, a(n) >= 0 iff n is in A000378.

If we maximize x we get A261904.

			Table showing initial values of n,x,y,z:
   0  0  0  0
   1  1  0  0
   2  1  1  0
   3  1  1  1
   4  2  0  0
   5  2  1  0
   6  2  1  1
   7 -1 -1 -1
   8  2  2  0
   9  2  2  1
  10  3  1  0
  11  3  1  1
  12  2  2  2
  13  3  2  0
  14  3  2  1
  15 -1 -1 -1
  16  4  0  0
  17  3  2  2
  18  3  3  0
  19  3  3  1
  20  4  2  0
  ...

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program
Index entries for sequences related to sums of squares

Cf. A000378, A004215, A005875, A261904.

Analogs for 4 squares: A178786 and A122921.

a(17) corrected, more terms from David Consiglio, Jr., Sep 11 2015

cp's OEIS Frontend

A034044 Numbers that are primitively but not imprimitively represented by x^2+y^2+z^2.

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A034046 Numbers that are both primitively and imprimitively represented by x^2+y^2+z^2.

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A072400 (Factors of 4 removed from n) modulo 8.

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A125084 Cubes which have a partition as the sum of 3 squares.

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A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

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Mathematica

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A004014 Norms of vectors in the b.c.c. lattice.

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Maple

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A124978 Smallest positive number which has exactly n different partitions as a sum of 4 squares x^2+y^2+z^2+t^2.

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Haskell

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A166687 Numbers of the form x^2 + y^2 + 1, x, y integers.

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