A263715 -id:A263715 - cp's OEIS frontend

A097269 Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.

2, 10, 18, 26, 34, 50, 58, 74, 82, 90, 98, 106, 122, 130, 146, 162, 170, 178, 194, 202, 218, 226, 234, 242, 250, 274, 290, 298, 306, 314, 338, 346, 362, 370, 386, 394, 410, 442, 450, 458, 466, 482, 490, 514, 522, 530, 538, 554, 562, 578, 586, 610, 626, 634

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and complement of A024352 (difference of squares).
Numbers of the form 4k+2 = double of an odd number, with the odd number equal to the sum of 2 squares (sequence A057653). - Jean-Christophe Hervé, Oct 24 2015
Numbers that are the sum of two odd squares. - Jean-Christophe Hervé, Oct 25 2015

			2 = 1^2 + 1^2, 10 = 1^2 + 3^2, 18 = 3^2 + 3^2.

Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097268, A097270, A097271. Equals twice A057653.

Cf. A020668, A263715.

is(n)=if(n%4!=2,return(0)); my(f=factor(n/2)); for(i=1,#f[,1],if(bitand(f[i,2],1)==1&&bitand(f[i,1],3)==3, return(0))); 1 \\ Charles R Greathouse IV, May 31 2013

from itertools import count, islice
from sympy import factorint
def A097269_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n//2).items()),count(2,4))
A097269_list = list(islice(A097269_gen(),30)) # Chai Wah Wu, Jun 28 2022

A062316 Neither the sum or difference of 2 squares.

Original entry on oeis.org

6, 14, 22, 30, 38, 42, 46, 54, 62, 66, 70, 78, 86, 94, 102, 110, 114, 118, 126, 134, 138, 142, 150, 154, 158, 166, 174, 182, 186, 190, 198, 206, 210, 214, 222, 230, 238, 246, 254, 258, 262, 266, 270, 278, 282, 286, 294, 302, 310, 318, 322, 326, 330, 334, 342, 350, 354, 358

Offset: 1

Michel ten Voorde, Jul 05 2001

nonn

Elements of A022544 congruent to 2 (mod 4).
Union of numbers congruent to 6 mod 8 (A017137) with numbers of the form 2 * A084109(n). - Franklin T. Adams-Watters, Jan 21 2007
Explanation: odd numbers are equal to the difference between two successive squares and among even numbers, multiples of 4 are of the form (k+2)^2-k^2, thus odd numbers and multiples of 4 are not in the sequence. Conversely, a difference of 2 squares cannot equal 2 (mod 4), thus this sequence contains the integers of the form 4k+2 that are in A022544 (not the sum of two squares); among integers of form 4k+2, this sequence contains all the integers of the form 8n+6 (A017137) that are not the sum of 2 squares because they have at least one prime factor congruent to 3 (mod 4) to an odd power; it also contains integers of the form 8n+2 = 2(4n+1) with 4n+1 not the sum of two squares, which is sequence A084109. - Jean-Christophe Hervé, Oct 24 2015

			From _Jean-Christophe Hervé_, Oct 24 2015: (Start)
6, 14, 22, 30, 38, 46, ... are in the sequence because they equal 6 (mod 8).
42 = 2*3*7, 66 = 2*3*11, 114 = 2*7*11 are also in the sequence: of the form 2*(4n+1) with 4n+1 not the sum of 2 squares.
(End)

Jean-Christophe Hervé, Table of n, a(n) for n = 1..2507

Cf. A022544, A016825, union of A017137 and 2*A084109, complement of A263715.

Cf. A097271, A079299.

N:= 1000: # to get all terms <= N
S:= {seq(4*i+2,i=0..floor((N-2)/4))}
  minus {seq(seq(x^2 + y^2, y = x .. floor(sqrt(N-x^2)),2),x=1..floor(sqrt(N)))}:
sort(convert(S,list)); # Robert Israel, Oct 25 2015

Select[Range@ 360, SquaresR[2, #] == 0 && Mod[#, 4] == 2 &] (* Michael De Vlieger, Oct 26 2015, after Harvey P. Dale at A022544 *)

a(n) == 2 (mod 4). Subsequence of A016825 (non-differences of squares). All first differences are either 4 or 8, each of which occurs infinitely often. - David W. Wilson, Mar 09 2005

Lim_{n->inf} a(n)/n = 4.

More terms from David W. Wilson, Feb 11 2003

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

A263765 Minimum number of squares necessary to write n as a sum or difference of squares.

Original entry on oeis.org

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

Offset: 0

Jean-Christophe Hervé, Oct 25 2015

nonn

This sequence is equivalent to A002828 (least number of squares that add up to n) for sums and differences of squares. Here the possible forms include not only sums of squares, but also differences like x^2 - y^2 or x^2 + y^2 - z^2.
a(n) <= A002828(n) which is <= 4 (Lagrange's "Four Squares theorem"). In fact, a(n) <= 3: numbers of the form 4k, 4k+1 or 4k+3 are equal to the difference of two squares, therefore a(n) <= 2 in this case, and a(4k+2) <= 3 because 4k+2 = 4k+1+1^2. More precisely, a(4k) = 1 or 2; a(4k+1) = 1 or 2; a(4k+2) = 2 or 3; a(4k+3) = 2.
If A002828(n) = 4, a(n) = 2 (see A004215); if A002828(n) = 3, a(n) = 2 or 3: this shows that the form x^2 + y^2 - z^2 is never necessary to write an integer with the minimum number of squares; and of course, if A002828 = 1 or 2, a(n) = A002828.

			a(6) = 3 because 6 = 2^2 + 1^2 + 1^2 and 6 is not the sum or the difference of two squares; a(28) = 2 because 28 = 8^2 - 6^2.

Jean-Christophe Hervé, Table of n, a(n) for n = 0..10000

Cf. A002828, A000290, A000415, A062316, A263715, A263737.

Using the partition of the natural numbers into A000290 (square numbers), A000415 (sum of 2 nonzero squares), A263737 (difference but not sum of 2 squares) and A062316 (neither the sum or difference of 2 squares), the sequence is completely defined by: a(A000290(n)) = 1, a(A000415(n)) = a(A263737(n)) = 2, a(A062316(n)) = 3.

cp's OEIS Frontend

A097269 Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A062316 Neither the sum or difference of 2 squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A263765 Minimum number of squares necessary to write n as a sum or difference of squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula