A022544 -id:A022544 - cp's OEIS frontend

A046712 From the Bruck-Ryser theorem: n == 1 or 2 (mod 4) which are not the sum of 2 squares.

6, 14, 21, 22, 30, 33, 38, 42, 46, 54, 57, 62, 66, 69, 70, 77, 78, 86, 93, 94, 102, 105, 110, 114, 118, 126, 129, 133, 134, 138, 141, 142, 150, 154, 158, 161, 165, 166, 174, 177, 182, 186, 189, 190, 198, 201, 206, 209, 210, 213, 214, 217, 222, 230, 237, 238

Offset: 1

N. J. A. Sloane

Intersection of A022544 and A046712. - Reinhard Zumkeller, Aug 16 2011

M. Hall, Jr., Combinatorial Theory, Wiley, New York, 1986, see Theorem 12.3.2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. H. Bruck and H. J. Ryser, The nonexistence of certain projective planes, Canad. J. Math., 1 (1949), 88-93.
Eric Weisstein's World of Mathematics, Bruck-Ryser-Chowla Theorem.
Wikipedia, Bruck-Ryser-Chowla theorem.
Index entries for sequences related to sums of squares

a046712 n = a046712_list !! (n-1)
a046712_list = filter ((`elem` [1,2]) . (`mod` 4)) a022544_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range[240], (Mod[#, 4] == 1 || Mod[#, 4] == 2) && PowersRepresentations[#, 2, 2] == {} & ] (* Jean-François Alcover, Aug 30 2011 *)
Select[Range[250],MemberQ[{1,2},Mod[#,4]]&&SquaresR[2,#]==0&] (* Harvey P. Dale, Apr 01 2015 *)

More terms from James Sellers

A111909 Numbers that cannot be represented as a^4 + b^2 with a, b > 0.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 84, 86, 87, 88

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

			3 cannot be represented as a^4 + b^2 and thus is in this sequence while 10 = 1^4 + 3^2 is not.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A111925 (complement), A055394, A022549. A022544 is a subsequence.

list(lim)=lim\=1; my(v=List(),u=vectorsmall(lim),m2); for(m=1,sqrtint(lim-1), m2=m^2; for(n=1,sqrtnint(lim-m2,4), u[m2+n^4]=1)); for(i=1,#u, if(!u[i], listput(v,i))); Set(v) \\ Charles R Greathouse IV, Sep 01 2015

a(n) = n + O(n^(3/4)). - Charles R Greathouse IV, Sep 01 2015

A119973 Numbers of the form (4k+1)*2^j which are not a sum of two squares.

Original entry on oeis.org

21, 33, 42, 57, 66, 69, 77, 84, 93, 105, 114, 129, 132, 133, 138, 141, 154, 161, 165, 168, 177, 186, 189, 201, 209, 210, 213, 217, 228, 237, 249, 253, 258, 264, 266, 273, 276, 282, 285, 297, 301, 308, 309, 321, 322, 329, 330, 336, 341, 345, 354, 357, 372

Offset: 1

Alford Arnold, Jun 03 2006

Intersection of A091072 and A022544. - Robert Israel, Oct 28 2018

			42 is there because it's (4*5+1)*2^1 and is not a sum of two squares.

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

Cf. A001481, A022544, A084109, A091072, A120772.

filter:= proc(n) local w; w:= n/2^padic:-ordp(n,2);
w mod 4 = 1 and select(t -> t[2]::odd and t[1] mod 4 = 3, ifactors(w)[2]) <> []
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 28 2018

okQ[n_] := EvenQ[(n/2^IntegerExponent[n, 2]-1)/2] && SquaresR[2, n] == 0;
Select[Range[1000], okQ] (* Jean-François Alcover, Feb 09 2023 *)

More terms from Don Reble, Jul 24 2006

A263737 Nonnegative integers that are the difference of two squares but not the sum of two squares.

Original entry on oeis.org

3, 7, 11, 12, 15, 19, 21, 23, 24, 27, 28, 31, 33, 35, 39, 43, 44, 47, 48, 51, 55, 56, 57, 59, 60, 63, 67, 69, 71, 75, 76, 77, 79, 83, 84, 87, 88, 91, 92, 93, 95, 96, 99, 103, 105, 107, 108, 111, 112, 115, 119, 120, 123, 124, 127, 129, 131, 132, 133, 135, 139, 140

Offset: 1

Jean-Christophe Hervé, Oct 25 2015

nonn

Intersection of A022544 (not the sum of two squares) and A042965 (differences of two squares).

The sequence contains all 4k + 3 and no 4k + 2 integers, and some 4k (4*A022544) and 4k+1 (A084109) integers. First differences are thus 1, 2, 3 or 4, each occurring infinitely often.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..5000

Cf. A001481, A022544, A042965, A016825, A020668, A062316, A097269, A263715.

rs[n_] := Reduce[n == x^2 + y^2, {x, y}, Integers]; rd[n_] := Reduce[0 <= y <= x && n == x^2 - y^2, {x, y}, Integers]; Reap[Do[If[rs[n] == False && rd[n] =!= False, Sow[n]], {n, 0, 140}]][[2, 1]] (* Jean-François Alcover, Oct 26 2015 *)

from itertools import count, islice
from sympy import factorint
def A263737_gen(): # generator of terms
    return filter(lambda n:n & 3 != 2 and any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()),count(0))
A263737_list = list(islice(A263737_gen(),30)) # Chai Wah Wu, Jun 28 2022

A343991 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are plane-filling but not perfect.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 8, 0, 4, 0, 12

Offset: 1

N. J. A. Sloane, May 06 2021

Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n-1 U's and D's, the Up and Down folds. These are also known as n-folds. In the square grid they uniquely correspond to folding morphisms, which are a special class of morphisms sigma on the alphabet {a,b,c,d}. (There is in particular the requirement that the first two letters of sigma(a) are ab.) Here the letters a,b,c, and d correspond to the four possible steps of the walk.
A curve C = C1 of order n generates curves Cj of order n^j by the process of iterated folding. Iterated folding corresponds to iterates of the folding morphism.
Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj.
Perfect means that four 90-degree rotated copies of the curves Cj started at the origin will pass exactly twice through all grid-points as j tends to infinity (except the origin itself).
It is a theorem that a(A022544(n)) = 0 for all n.

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted and updated in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614.

Michel Dekking, Table of n, a(n) for n = 1..20
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
F. M. Dekking, Paperfolding Morphisms, Planefilling Curves, and Fractal Tiles, Theoretical Computer Science, volume 414, issue 1, January 2012, pages 20-37. Also arXiv:1011.5788 [math.CO], 2010-2011.

Cf. A296148, A343990, A343992.

Rewritten and renamed by Michel Dekking, Jun 06 2021

A353219 Positive integers which cannot be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

Original entry on oeis.org

7, 12, 15, 22, 23, 24, 28, 31, 39, 43, 44, 47, 48, 55, 56, 57, 60, 63, 67, 70, 71, 76, 77, 78, 79, 84, 87, 88, 92, 93, 94, 95, 96, 103, 108, 111, 112, 115, 119, 120, 124, 127, 132, 133, 134, 135, 139, 140, 141, 142, 143, 151, 152, 155, 156, 159, 166, 167, 168, 172, 175

Offset: 1

Eric Fox, Apr 30 2022

nonn

These are the numbers which cannot be a clue in a Tasquare puzzle.

Tasquare also known as Tasukuea.

			A 9 clue can be satisfied in at least one way:
  OOO  OO OO
  OOO  OO9OO
  OOO9   O
So, 9 is not a term in this sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Cross+A, Tasquare rules
Nikoli, Tasukuea rules [broken link]

Complement of A353202.

Cf. A022544.

q:= proc(n) local i; for i to isqrt(n) do if ormap(issqr,
      [n-i^2, n-i^2-1]) then return false fi od: true
    end:
select(q, [$1..175])[];  # Alois P. Heinz, Apr 30 2022

q[n_] := Module[{i}, For[i = 1, i <= Sqrt[n], i++, If[AnyTrue[ {n-i^2, n-i^2-1}, IntegerQ@Sqrt[#]&], Return[False]]]; True];
Select[Range[175], q] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)

A383785 Numbers not occurring as norms of vectors in any regular planar lattice.

Original entry on oeis.org

6, 11, 14, 15, 22, 23, 24, 30, 33, 35, 38, 42, 44, 46, 47, 51, 54, 55, 56, 59, 60, 62, 66, 69, 70, 71, 77, 78, 83, 86, 87, 88, 92, 94, 95, 96, 99, 102, 105, 107, 110, 114, 115, 118, 119, 120, 123, 126, 131, 132, 134, 135, 138, 140, 141, 142, 143, 150, 152, 154

Offset: 1

C. S. Davis, May 09 2025

nonn

The norms of Hurwitz quaternions strictly noncoplanar with the origin and any nonopposing pair of unit Hurwitz quaternions, due to Lagrange's four-square theorem and the complement's representation of every coplanar Hurwitz quaternion norm. - C. S. Davis, May 19 2025

C. S. Davis, Table of n, a(n) for n = 1..7374

Intersection of A022544 and A034020.
Complement of A383784.
Supersequence of A055039.

A034024 Imprimitively but not primitively represented by x^2+y^2.

Original entry on oeis.org

4, 8, 9, 16, 18, 20, 32, 36, 40, 45, 49, 52, 64, 68, 72, 80, 81, 90, 98, 100, 104, 116, 117, 121, 128, 136, 144, 148, 153, 160, 162, 164, 180, 196, 200, 208, 212, 225, 232, 234, 242, 244, 245, 256, 260, 261, 272, 288, 292, 296, 306, 320, 324, 328

Offset: 0

N. J. A. Sloane

nonn

Cf. A001481, A008784, A022544, A034023-.

A125111 Cubes which do not have a partition as the sum of 2 squares.

Original entry on oeis.org

27, 216, 343, 1331, 1728, 2744, 3375, 6859, 9261, 10648, 12167, 13824, 19683, 21952, 27000, 29791, 35937, 42875, 54872, 59319, 74088, 79507, 85184, 97336, 103823, 110592, 132651, 157464, 166375, 175616, 185193, 205379, 216000, 238328, 250047

Offset: 1

Artur Jasinski, Nov 21 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000578, A022544, A125084, A125110.

Select[Range[0, 65]^3, SquaresR[2, # ] == 0 &] (* Ray Chandler, Nov 23 2006 *)

isA125111(ncube)={ local(a) ; a=0; while(a^2<=ncube, if(issquare(ncube-a^2), return(0) ; ) ; a++ ; ) ; return(1) ; } { for(n=0,200, if(isA125111(n^3), print1(n^3,",") ; ) ; ) ; } \\ R. J. Mathar, Nov 23 2006

Equals A000578 INTERSECT A022544. - R. J. Mathar, Nov 23 2006

a(n) = A022544(n)^3. - Ray Chandler, Nov 23 2006

More terms from R. J. Mathar and Ray Chandler, Nov 23 2006

A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

Original entry on oeis.org

34, 136, 146, 178, 194, 205, 221, 305, 306, 377, 386, 410, 466, 482, 505, 514, 544, 545, 562, 584, 674, 689, 706, 712, 745, 776, 793, 802, 820, 850, 866, 884, 890, 898, 905, 1154, 1186, 1202, 1205, 1220, 1224, 1234, 1282, 1314, 1345, 1346, 1394, 1405, 1469

Offset: 1

Max Alekseyev, Jan 21 2010

nonn

If the fundamental unit y0 + x0*sqrt(n) of Q(sqrt(n)) has norm -1, then (x0,y0) represents a rational solution to Pell equation y^2 - n*x^2 = -1. For n in this sequence, rational solutions exist but not delivered by the fundamental unit.

Set difference of A000415 and its subsequence A172000.
Set difference of A087643 and its subsequence A022544.
Squarefree terms form A031398.
Odd terms form A249052.
Cf. A031399, A137351.

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398.

Edited by Max Alekseyev, Mar 09 2010

cp's OEIS Frontend

A046712 From the Bruck-Ryser theorem: n == 1 or 2 (mod 4) which are not the sum of 2 squares.

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A111909 Numbers that cannot be represented as a^4 + b^2 with a, b > 0.

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A119973 Numbers of the form (4k+1)*2^j which are not a sum of two squares.

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Maple

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A263737 Nonnegative integers that are the difference of two squares but not the sum of two squares.

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A343991 Number of grid-filling curves of order n (on the square grid) with turns by +-90 degrees generated by folding morphisms that are plane-filling but not perfect.

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Extensions

A353219 Positive integers which cannot be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

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Examples

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Programs

Maple

Mathematica

A383785 Numbers not occurring as norms of vectors in any regular planar lattice.

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A034024 Imprimitively but not primitively represented by x^2+y^2.

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A125111 Cubes which do not have a partition as the sum of 2 squares.

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