A005767 -id:A005767 - cp's OEIS frontend

A094958 Numbers of the form 2^k or 5*2^k.

1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152

Offset: 1

Ralf Stephan, Jun 01 2004

The subset {a(1),...,a(2k)} together with a(2k+2) is the set of proper divisors of 5*2^k.
For a(n)>4: number of vertices of complete graphs that can be properly edge-colored in such a way that the edges can be partitioned into edge disjoint multicolored isomorphic spanning trees.
(Editor's note: The following 3 comments are equivalent.)
From Wouter Meeussen, Apr 10 2005: This appears to be the same sequence as "Numbers n such that n^2 is not the sum of three nonzero squares". Don Reble and Paul Pollack respond: Yes, that is correct.
Also numbers k such that k^2=a^2+b^2+c^2 has no solutions in the positive integers a, b and c. - Wouter Meeussen, Apr 20 2005
The only natural numbers which cannot be the lengths of an interior diagonal of a cuboid with natural edges. - Michael Somos, Mar 02 2004

Wacław Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 101, MR2002669.

Gregory Constantine, Multicolored isomorphic spanning trees in complete graphs, Discrete Mathematics and Theoretical Computer Science, Vol. 5 (2002), pp. 121-126.
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A029744, A029745.
Union of A000079 and A020714.
Complement of A005767.

With[{c=2^Range[0,30]},Union[Join[c,5c]]] (* Harvey P. Dale, Jul 15 2012 *)

def A094958(n): return 1<>1)+1 if n&1 else 5<<((n>>1)-2) # Chai Wah Wu, Feb 14 2025

a(1)=1, a(2)=2, a(3)=4, for n>=0, a(2n+3) = 4*2^n, a(2n+4) = 5*2^n.
Recurrence: for n>4, a(n) = 2a(n-2).
G.f.: x*(1+x)*(1+x+x^2)/(1-2x^2).
Sum_{n>=1} 1/a(n) = 12/5. - Amiram Eldar, Jan 21 2022

Edited by T. D. Noe and M. F. Hasler, Nov 12 2010

A169580 Squares of the form x^2+y^2+z^2 with x,y,z positive integers.

Original entry on oeis.org

9, 36, 49, 81, 121, 144, 169, 196, 225, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364

Offset: 1

N. J. A. Sloane, Mar 02 2010

nonn

Integer solutions of a^2 = b^2 + c^2 + d^2, i.e., Pythagorean Quadruples. - Jon Perry, Oct 06 2012

Also null (or light-like, or isotropic) vectors in Minkowski 4-space. - Jon Perry, Oct 06 2012

			9 = 1 + 4 + 4,
36 = 16 + 16 + 4,
49 = 36 + 9 + 4,
81 = 49 + 16 + 16,
so these are in the sequence.
16 cannot be written as the sum of 3 squares if zero is not allowed, therefore 16 is not in the sequence.
Also we can see that 49-36-9-4=0, so (7,6,3,2) is a null vector in the signatures (+,-,-,-) and (-,+,+,+). - _Jon Perry_, Oct 06 2012

T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

Robert Israel, Table of n, a(n) for n = 1..3140
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.
D. Rabahy, Spreadsheet revealing sequence in table of all a^2+b^2+c^2
David Rabahy, a^2+b^2+c^2 squares colored, zoomed way out to make "lines" apparent

For the square roots see A005767. Cf. A000378, A000419.

Cf. A217554.

M:= 10000: # to get all terms <= M
sort(convert(select(issqr, {seq(seq(seq(x^2 + y^2 + z^2,
  z=y..floor(sqrt(M-x^2-y^2))), y=x..floor(sqrt((M-x^2)/2))),
x=1..floor(sqrt(M/3)))}),list)); # Robert Israel, Jan 28 2016

Select[Range[60]^2, Resolve@ Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers], And[x > 0, y > 0, z > 0]] &] (* Michael De Vlieger, Jan 27 2016 *)

A005818 Numbers n that are primitive solutions to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

Original entry on oeis.org

3, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

N. J. A. Sloane, RALPH PETERSON (ralphp(AT)LIBRARY.NRL.NAVY.MIL)

These are the odd numbers excluding 1 and 5. Every term in A005767 is n*2^k for some k.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005767.

Join[{3}, Range[7, 200, 2]] (* Paolo Xausa, Feb 14 2025 *)

def A005818(n): return (n<<1)+3 if n>1 else 3 # Chai Wah Wu, Feb 14 2025

G.f.: x*(-2*x^2 + x + 3)/(x - 1)^2. - Chai Wah Wu, Sep 14 2018

E.g.f.: (exp(x) - 1)*(3 + 2*x). - Stefano Spezia, Feb 19 2023

Name corrected by T. D. Noe, Nov 12 2010

A118901 Volumes of cuboids with integer sides and main diagonal.

Original entry on oeis.org

4, 32, 36, 108, 112, 140, 144, 220, 252, 256, 288, 364, 396, 400, 500, 540, 608, 612, 644, 756, 832, 864, 896, 900, 972, 1012, 1120, 1152, 1292, 1364, 1372, 1404, 1408, 1508, 1620, 1728, 1760, 1764, 1872, 1904, 1980, 1984

Offset: 1

Giovanni Resta, May 05 2006

nonn

			32 is the volume of the cuboid of sides 8,4,1 which has main diagonal = 9.

Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..100 from Harvey P. Dale).
Eric Weisstein's World of Mathematics, Cuboid.

Cf. A005767, A118899, A118902.

With[{nn=50},Take[Union[Times@@@Select[Union[Sort/@Tuples[Range[ 2nn],3]], IntegerQ[ Sqrt[ Total[#^2] ]]&]],nn]] (* Harvey P. Dale, Jun 05 2016 *)

A118902 Surfaces of cuboids with integer sides and main diagonal.

Original entry on oeis.org

16, 64, 72, 88, 144, 168, 192, 216, 240, 256, 264, 288, 304, 336, 352, 400, 432, 480, 496, 520, 552, 576, 592, 600, 640, 648, 672, 696, 720, 760, 768, 784, 792, 840, 864, 960, 1008, 1024, 1056, 1120, 1152, 1176, 1216, 1224, 1248, 1296, 1312, 1320, 1344, 1368, 1408, 1440, 1464, 1504

Offset: 1

Giovanni Resta, May 05 2006

nonn

			88 is the surface of the cuboid of sides 8,4,1 which has main diagonal = 9.

Robin Visser, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cuboid.

Cf. A005767, A118899, A118901.

More terms from Robin Visser, Jan 02 2024

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

A341329 Numbers k such that k^2 is the sum of m nonzero squares for all 1 <= m <= k^2 - 14.

Original entry on oeis.org

13, 15, 17, 25, 26, 29, 30, 34, 35, 37, 39, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137

Offset: 1

Jianing Song, Feb 09 2021

Numbers k such that k^2 is in A018820. Note that k^2 is never the sum of k^2 - 13 positive squares.
A square k^2 is the sum of m positive squares for all 1 <= m <= k^2 - 14 if k^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma).
Intersection of A009003 and A005767. Also A009003 \ A020714.
Numbers k not of the form 5*2^e such that k has at least one prime factor congruent to 1 modulo 4.
Has density 1 over all positive integers.

			13 is a term: 169 = 13^2 = 5^2 + 12^2 = 3^2 + 4^2 + 12^2 = 11^2 + 4^2 + 4^2 + 4^2 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2 = 6^2 + 6^2 + 6^2 + 6^2 + 4^2 + 3^2 = ... = 3^2 + 2^2 + 2^2 + 1^2 + 1^2 + ... + 1^2 (sum of 155 positive squares, with 152 (1^2)'s), but 169 cannot be represented as the sum of 156 positive squares.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A018820, A309778, A009003, A005767, A020714.

isA341329(n) = setsearch(Set(factor(n)[, 1]%4), 1) && !(n/5 == 2^valuation(n, 2))

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 *)

A178096 Cube of n is equal to sum of four positive distinct squares; n^3=a^2+b^2+c^2+d^2; a>b>c>d>0.

Original entry on oeis.org

5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 19 2010

nonn

5^3=8^2+6^2+4^2+3^2, 6^3=10^2+8^2+6^2+4^2, ...

Cf. A003294, A005767, A023809, A046100.

z=100;lst={};Do[a2=a^2;Do[b2=b^2;Do[c2=c^2;Do[d2=d^2;e2=a2+b2+c2+d2;e=e2^(1/3);If[IntegerQ[e],AppendTo[lst,e]],{d,c-1,1,-1}],{c,b-1,1,-1}],{b,a-1,1,-1}],{a,1,z}];Union@lst

{n: n^3 in A004433}. - R. J. Mathar, Jun 15 2018

Terms > 33 from R. J. Mathar, Jun 15 2018

A274255 Numbers n such that n^2 is the sum of three nonzero squares while n is not.

Original entry on oeis.org

7, 13, 15, 23, 25, 28, 31, 37, 39, 47, 52, 55, 58, 60, 63, 71, 79, 85, 87, 92, 95, 100, 103, 111, 112, 119, 124, 127, 130, 135, 143, 148, 151, 156, 159, 167, 175, 183, 188, 191, 199, 207, 208, 215, 220, 223, 231, 232, 239, 240, 247, 252, 255, 263, 271, 279, 284

Offset: 1

Altug Alkan, Jun 16 2016

			7 is a term because 7 is not in A000408 and 7^2 = 49 = 2^2 + 3^2 + 6^2.

Cf. A000408, A005767.

isA000408(n) = my(a, b) ; a=1 ; while(a^2+1A000408(n) && isA000408(n^2), print1(n, ", ")));

cp's OEIS Frontend

A094958 Numbers of the form 2^k or 5*2^k.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A169580 Squares of the form x^2+y^2+z^2 with x,y,z positive integers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

A005818 Numbers n that are primitive solutions to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A118901 Volumes of cuboids with integer sides and main diagonal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A118902 Surfaces of cuboids with integer sides and main diagonal.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

A341329 Numbers k such that k^2 is the sum of m nonzero squares for all 1 <= m <= k^2 - 14.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

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

Views