A097269 -id:A097269 - cp's OEIS frontend

A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

5, 8, 13, 17, 20, 25, 29, 32, 37, 40, 41, 45, 52, 53, 61, 65, 68, 72, 73, 80, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 125, 128, 136, 137, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 197, 200, 205, 208, 212, 221, 225, 229, 232, 233

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and A024352 (difference of squares).

Also: Numbers of the form x^2+4y^2, where x and y are positive integers. Cf. A154777, A092572, A154778 for analogous sequences. - M. F. Hasler, Jan 24 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097269, A097270, A097271.

isA097268(n) = forstep( b=2,sqrtint(n-1),2, issquare(n-b^2) && return(1)) \\ M. F. Hasler, Jan 24 2009

A020668 Numbers of the form x^2 + 4*y^2.

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 13, 16, 17, 20, 25, 29, 32, 36, 37, 40, 41, 45, 49, 52, 53, 61, 64, 65, 68, 72, 73, 80, 81, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 121, 125, 128, 136, 137, 144, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 196, 197, 200, 205, 208

Offset: 1

David W. Wilson

x^2 + 4y^2 has discriminant -16.
Numbers that can be expressed as both the sum of two squares and the difference of two squares; the intersection of sequences A001481 and A042965. - T. D. Noe, Feb 05 2003
A004531(n) is nonzero if and only if n is of the form x^2 + 4*y^2. - Michael Somos, Jan 05 2012
These are the sum of two squares that are congruent to 0 or 1 (mod 4), and thus that are also the difference of two squares. - Jean-Christophe Hervé, Oct 25 2015

Jean-Christophe Hervé, Table of n, a(n) for n = 1..7500
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A001481, A004531, A042965, A097269. For primes see A002144.

[n: n in [0..208] | NormEquation(4, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

Select[Range[0, 300], SquaresR[2, #] != 0 && Mod[#, 4] != 2&] (* Jean-François Alcover, May 13 2017 *)

for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0, print1(n, ", "))) \\ Altug Alkan, Oct 29 2015

Complement of A097269 in A001481. - Jean-Christophe Hervé, Oct 25 2015

A057653 Odd numbers of form x^2 + y^2.

Original entry on oeis.org

1, 5, 9, 13, 17, 25, 29, 37, 41, 45, 49, 53, 61, 65, 73, 81, 85, 89, 97, 101, 109, 113, 117, 121, 125, 137, 145, 149, 153, 157, 169, 173, 181, 185, 193, 197, 205, 221, 225, 229, 233, 241, 245, 257, 261, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 333, 337

Offset: 1

N. J. A. Sloane, Oct 15 2000

nonn

Numbers with only odd prime factors and such that all prime factors congruent to 3 modulo 4 occur to an even exponent. - Jean-Christophe Hervé, Oct 24 2015
Odd terms of A020668. - Altug Alkan, Nov 19 2015
Also one half of the numbers that are the sum of two odd squares (without multiplicity). See A097269 for twice the numbers. - Wolfdieter Lang, Jan 12 2017

Jean-Christophe Hervé, Table of n, a(n) for n = 1..4000
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).

Odd members of A001481.
Odd members of A020668.
Complement of A084109 in 4k+1 numbers (A016813).
Cf. A016754 (odd squares), A097269.

readlib(issqr): for n from 1 to 1001 by 2 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d,`,n); break fi; od:od:

fQ[n_] := Length@ Catch@ Do[If[IntegerQ@ Sqrt[n - k^2], Throw[{k, Sqrt[n - k^2]}], Nothing], {k, Floor[Sqrt@ n]^2}] != 0; Select[Range[1, 340, 2], fQ] (* Michael De Vlieger, Nov 13 2015 *)

isok(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
for(n=1, 1e3, if(isok(n) && n%2==1, print1(n", "))) \\ Altug Alkan, Nov 13 2015

for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0 && n%2==1, print1(n, ", "))) \\ Altug Alkan, Nov 19 2015

from itertools import count, islice
from sympy import factorint
def A057653_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(1,2))
A057653_list = list(islice(A057653_gen(),30)) # Chai Wah Wu, Jun 28 2022

n = odd square * {product of distinct primes == 1 (mod 4)}.

a(n) = A097269(n)/2. - Wolfdieter Lang, Jan 12 2017

More terms from James Sellers, Oct 16 2000

A101412 Least number of odd squares that sum to n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Aug 08 2009

			a(13) = 5: 13 = 1+1+1+1+9.

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

Cf. A002828, A151925, A097269.

A101412 := proc(n) local lsq; lsq := [seq((2*j+1)^2,j=0..floor((sqrt(n)-1)/2))] ; lsq := convert(lsq,set) ; a := n ; for p in combinat[partition](n) do if convert(p,set) minus lsq = {} then a := min(a,nops(p)) ; fi; od: a ; end: for n from 1 do printf("%d,\n",A101412(n)) ; od: # R. J. Mathar, Aug 08 2009
# problem has optimal substructure:
a:= proc(n) option remember; local r; r:= isqrt(n);
      `if`(r^2=n and irem(r, 2)=1, 1,
       min(seq(a(i)+a(n-i), i=1..n/2)))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 31 2011

a[n_] := a[n] = Module[{r}, r = Sqrt[n]; If[IntegerQ[r] && OddQ[r], 1, Min[Table[a[i]+a[n-i], {i, 1, Floor[n/2]}]]]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)

a(n)={x=n-1;if(x%8>1,k=1+x%8);if(n%8==1,k=9;if(issquare(n)&&n%2==1,k=1));if(x%8==1,k=10;y=1;while(x>0,if(issquare(x)&&x%2==1,k=2);y=y+2;x=n-y^2));k;} \\ Jinyuan Wang, Jan 29 2019

From Jinyuan Wang, Jan 29 2019: (Start)
For n == 1 (mod 8), if n is a perfect square, a(n) = 1, otherwise a(n) = 9.
For n == 2 (mod 8), if n is a term in A097269, a(n) = 2, otherwise a(n) = 10.
For n == k (mod 8), k = 3,4,...,8, a(n) = k.
For positive integer x, a(72*x+42) = a(72*x+66) = 10. (End)

More terms from R. J. Mathar, Aug 08 2009

More terms from Alois P. Heinz, Jan 30 2011

A263715 Nonnegative integers that are the sum or difference of two squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80

Offset: 1

Jean-Christophe Hervé, Oct 24 2015

nonn

Contains all integers that are not equal to 2 (mod 4) (they are of the form y^2 - x^2) and those of the form 4k+2 = 2*(2k+1) with the odd number 2k+1 equal to the sum of two squares (A057653).

			2 = 1^2 + 1^2, 3 = 2^2 - 1^2, 4 = 2^2 + 0^2, 5 = 2^2 + 1^2 = 3^2 - 2^2.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Cf. A001481, A057653, A097269, A042965, A020668, A263737.

Cf. A062316 (complement), A079298.

r[n_] := Reduce[n == x^2 + y^2, {x, y}, Integers] || Reduce[0 <= y <= x && n == x^2 - y^2, {x, y}, Integers]; Reap[Do[If[r[n] =!= False, Sow[n]], {n, 0, 80}]][[2, 1]] (* Jean-François Alcover, Oct 25 2015 *)

from itertools import count, islice
from sympy import factorint
def A263715_gen(): # generator of terms
    return filter(lambda n: n & 3 != 2 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A263715_list = list(islice(A263715_gen(),30)) # Chai Wah Wu, Jun 28 2022

Union of A001481 (sums of two squares) and A042965 (differences of two squares).
Union of A042965 and 2*A057653 = A097269, with intersection of A042965 and A097269 = {}.
Union of A020668 (x^2+y^2 and a^2-b^2), A097269 (x^2+y^2, not a^2-b^2) and A263737 (not x^2+y^2, a^2-b^2).

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

A096315 Dimensions n such that the integer lattice Z^n contains n+1 equidistant points (i.e., the vertices of a regular n-simplex).

Original entry on oeis.org

1, 3, 7, 8, 9, 11, 15, 17, 19, 23, 24, 25, 27, 31, 33, 35, 39, 43, 47, 48, 49, 51, 55, 57, 59, 63, 67, 71, 73, 75, 79, 80, 81, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 115, 119, 120, 121, 123, 127, 129, 131, 135, 139, 143, 145, 147, 151, 155, 159, 161, 163, 167

Offset: 1

David Radcliffe, Aug 01 2004

Schoenberg proved that a regular n-simplex can be inscribed in Z^n in the following cases and no others: (1) n is even and n+1 is a square; (2) n == 3 (mod 4); (3) n == 1 (mod 4) and n+1 is the sum of two squares.

			There is no equilateral triangle in the plane whose vertices have integer coordinates, so 2 is not on the list. But there is a regular tetrahedron in space whose vertices have integer coordinates, namely (0,0,0), (0,1,1), (1,0,1), (1,1,0), hence 3 is on the list.

Munenori Inagaki, Hideki Matsumura, Masanori Sawa, and Yukihiro Uchida, On a group of normalized solutions of the higher-dimensional Prouhet-Tarry-Escott problem, arXiv:2508.20733 [math.NT], 2025. See p. 6.
Hiroshi Maehara and Horst Martini, Elementary geometry on the integer lattice, Aequationes mathematicae, 92 (2018), 763-800. See Sec. 3.2.
I. J. Schoenberg, Regular Simplices and Quadratic Forms, J. London Math. Soc. 12 (1937) 48-55.

Contains A033996 except 0.

Cf. A000290, A001481, A016813, A097269.

select(n->(is(n,even) and issqr(n+1)) or (n mod 4 = 3) or ((n mod 4 = 1) and (numtheory[sum2sqr](n+1)<>[])),[ $1..200]);

A287960 Numbers that are the sum of two centered triangular numbers (A005448).

Original entry on oeis.org

2, 5, 8, 11, 14, 20, 23, 29, 32, 35, 38, 41, 47, 50, 56, 62, 65, 68, 74, 77, 83, 86, 89, 92, 95, 104, 110, 113, 116, 119, 128, 131, 137, 140, 146, 149, 155, 167, 170, 173, 176, 182, 185, 194, 197, 200, 203, 209, 212, 218, 221, 230, 236, 239, 245, 251, 254, 263, 266, 272, 275, 278, 281, 284, 293, 299

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A020756, A051533, A102795, A286636.

nmax = 300; f[x_] := Sum[x^(3 k (k - 1)/2 + 1), {k, 1, 20}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

8*a(n) = 10+3*A097269(n). - R. J. Mathar, Jul 26 2017

A339952 Numbers that are the sum of an even square > 0 and an odd square.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 45, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 117, 125, 137, 145, 149, 153, 157, 169, 173, 181, 185, 193, 197, 205, 221, 225, 229, 233, 241, 245, 257, 261, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 333, 337, 349, 353, 365, 369, 373, 377

Offset: 1

Wesley Ivan Hurt, Dec 24 2020

			13 is in the sequence since it is the sum of an even square > 0 and an odd square, 2^2 + 3^2 = 4 + 9 = 13.

Cf. A000404, A010052, A057653, A097269.

Table[If[Sum[Sign[(Mod[i, 2] Mod[n - i + 1, 2] + Mod[i + 1, 2] Mod[n - i, 2])] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 500}] // Flatten

def aupto(limit):
  m = int(limit**.5) + 2
  es = [i*i for i in range(2, m, 2)]
  os = [i*i for i in range(1, m, 2)]
  return sorted(set(a+b for a in es for b in os if a+b <= limit))
print(aupto(377)) # Michael S. Branicky, May 13 2021

A290275 Numbers that are the sum of distinct odd positive squares.

Original entry on oeis.org

1, 9, 10, 25, 26, 34, 35, 49, 50, 58, 59, 74, 75, 81, 82, 83, 84, 90, 91, 106, 107, 115, 116, 121, 122, 130, 131, 139, 140, 146, 147, 155, 156, 164, 165, 169, 170, 171, 178, 179, 180, 194, 195, 196, 202, 203, 204, 205, 211, 212, 218, 219, 225, 226, 227, 228, 234, 235, 236, 237, 243, 244, 250, 251, 252, 253

Offset: 1

Ilya Gutkovskiy, Jul 25 2017

nonn

Complement of A167703.

1922 is the largest of positive integers not in this sequence.

			139 is in the sequence because 139 = 9 + 49 + 81 = 3^2 + 7^2 + 9^2.

Cf. A003995, A016754, A097269, A167700, A167703.

max = 253; f[x_] := Product[1 + x^(2 k + 1)^2, {k, 0, 10}]; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, max}]] // Rest

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A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

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A020668 Numbers of the form x^2 + 4*y^2.

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A057653 Odd numbers of form x^2 + y^2.

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A101412 Least number of odd squares that sum to n.

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A263715 Nonnegative integers that are the sum or difference of two squares.

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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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A096315 Dimensions n such that the integer lattice Z^n contains n+1 equidistant points (i.e., the vertices of a regular n-simplex).

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