A022544 -id:A022544 - cp's OEIS frontend

A071011 Numbers n such that n is a sum of 2 squares (i.e., n is in A001481(k)) and sigma(n) == 0 (mod 4).

65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 340, 365, 370, 377, 410, 442, 445, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 580, 585, 610, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 820, 865, 884, 890, 901, 905

Offset: 1

Benoit Cloitre, May 19 2002

It is conjectured that if m is not a sum of 2 squares (i.e., m is in A022544(k)) sigma(m) == 0 (mod 4).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001481, A022544.

Select[Range[10^3], And[SquaresR[2, #] > 0, Divisible[DivisorSigma[1, #], 4]] &] (* Michael De Vlieger, Jul 30 2017 *)

for(n=1,1000,if(1-sign(sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))))+sigma(n)%4==0,print1(n,",")))

from math import prod
from itertools import count, islice
from sympy import factorint
def A071011_gen(): # generator of terms
    return filter(lambda n:(lambda f:all(p & 3 != 3 or e & 1 == 0 for p, e in f) and prod((p**(e+1)-1)//(p-1) & 3 for p, e in f) & 3 == 0)(factorint(n).items()),count(0))
A071011_list = list(islice(A071011_gen(),30)) # Chai Wah Wu, Jun 27 2022

A124981 Odd numbers that are not the sum of 2 squares.

Original entry on oeis.org

3, 7, 11, 15, 19, 21, 23, 27, 31, 33, 35, 39, 43, 47, 51, 55, 57, 59, 63, 67, 69, 71, 75, 77, 79, 83, 87, 91, 93, 95, 99, 103, 105, 107, 111, 115, 119, 123, 127, 129, 131, 133, 135, 139, 141, 143, 147, 151, 155, 159, 161, 163, 165, 167, 171, 175, 177, 179, 183, 187

Offset: 1

Artur Jasinski, Nov 15 2006

nonn

			13 is not in the list because it can be written as 3^2+2^2.

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

Cf. A005408, A022544.

Select[2 * Range[0, 100] + 1, SquaresR[2, #] == 0 &] (* Amiram Eldar, Mar 12 2020 *)

isA022544(n)={ local(cnt=0,y2) ; for(x=0,floor(sqrt(n)), y2=n-x^2 ; if( y2>=x^2 && issquare(y2), return(0) ; ) ; ) ; return(1) ; } isA124981(n)={ return( (n%2) && isA022544(n)) ; } { for(n=1,200, if( isA124981(n), print1(n,", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 29 2006

Equals A022544 INTERSECT A005408. - R. J. Mathar, Nov 29 2006

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

A173256 Partial sums of A001481.

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 29, 39, 52, 68, 85, 103, 123, 148, 174, 203, 235, 269, 305, 342, 382, 423, 468, 517, 567, 619, 672, 730, 791, 855, 920, 988, 1060, 1133, 1207, 1287, 1368, 1450, 1535, 1624, 1714, 1811, 1909, 2009, 2110, 2214, 2320, 2429, 2542, 2658, 2775

Offset: 1

Jonathan Vos Post, Feb 14 2010

nonn

The subsequence of primes in this sequence begins 3, 7, 29, 103, 269, 619, 1811, 3271.

			a(66) = 0 + 1 + 2 + 4 + 5 + 8 + 9 + 10 + 13 + 16 + 17 + 18 + 20 + 25 + 26 + 29 + 32 + 34 + 36 + 37 + 40 + 41 + 45 + 49 + 50 + 52 + 53 + 58 + 61 + 64 + 65 + 68 + 72 + 73 + 74 + 80 + 81 + 82 + 85 + 89 + 90 + 97 + 98 + 100 + 101 + 104 + 106 + 109 + 113 + 116 + 117 + 121 + 122 + 125 + 128 + 130 + 136 + 137 + 144 + 145 + 146 + 148 + 149 + 153 + 157 + 160 = 4876.

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

Cf. A001481, A022544, A004018, A000161, A002654, A064533, A000404, A002828, A000378, A025284-A025320, A125110, A091072.

N:= 1000:
A001481:= sort(convert({seq(seq(x^2+y^2, y=0..floor(sqrt(N-x^2))),x=0..floor(sqrt(N)))},list)):
ListTools:-PartialSums(A001481); # Robert Israel, Mar 15 2016

from itertools import count, accumulate, islice
from sympy import factorint
def A173256_gen(): # generator of terms
    return accumulate(filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0)))
A173256_list = list(islice(A173256_gen(),30)) # Chai Wah Wu, Jun 27 2022

a(n) = Sum_{i=1..n} A001481(i) = Sum_{i=1..n} (numbers that are the sum of 2 nonnegative squares) = Sum_{i=1..n} (numbers n such that i = x^2 + y^2 has a solution in nonnegative integers x, y).

a(21) corrected by Robert Israel, Mar 15 2016

A294716 Numbers that are the sum of 2 squares in exactly 5 ways.

Original entry on oeis.org

4225, 7225, 8125, 8450, 10625, 14450, 16250, 16900, 18125, 21025, 21250, 23125, 25625, 28900, 32500, 33125, 33800, 34225, 36250, 38025, 38125, 42025, 42050, 42500, 45625, 46250, 48841, 51250, 55625, 57800, 60625, 63125, 65000, 65025, 66250, 67600, 68125

Offset: 1

Robert Price, Nov 07 2017

nonn

Robert Price, Table of n, a(n) for n = 1..103
Index entries for sequences related to sums of squares

Cf. A000443, A022544.

Select[Range[10000], Length[PowersRepresentations[#, 2, 2]] == 5 &]

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

A357018 Records in the number of consecutive integers not expressible as sums of 2 squares.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 14, 18, 19, 20, 23, 24, 27, 28, 30, 33, 34, 35, 36, 47, 48, 49, 52, 55, 59, 60, 62, 63, 65, 67, 70, 71, 73, 79, 80, 81, 86, 87, 104, 106

Offset: 1

Hugo Pfoertner, Sep 09 2022

			  n  A297350(n) a(n)  Terms in A022544
  1     0        0    {}
  2     2        1    {3}
  3     5        2    {6, 7}
  4    20        4    {21, 22, 23, 24}
  5    74        5    {75, 76, 77, 78, 79}
  6    90        6    { 91, ...,  96}
  7   185        7    {186, ..., 192}
  8   377        8    {378, ..., 385}
  9   986       10    {987, ..., 996}

Cf. A001481, A022544, A104271, A260157, A297350.

A358311 Lucas numbers that are not the sum of two squares.

Original entry on oeis.org

3, 7, 11, 47, 76, 123, 199, 322, 843, 1364, 2207, 3571, 5778, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 4870847, 7881196, 12752043, 20633239, 33385282, 87403803, 141422324, 228826127, 370248451, 599074578, 1568397607, 2537720636

Offset: 1

Chai Wah Wu, Jan 10 2023

nonn

Lucas numbers with indices 2, 4, 5 mod 6 are 3 mod 4, so these are all terms. - Charles R Greathouse IV, Jan 11 2023

Chai Wah Wu, Table of n, a(n) for n = 1..958

Intersection of A000032 and A022544.

Cf. A356809.

R:= NULL: count:= 0:
a:= 2: b:= 1:
for i from 1 while count < 100 do
  a, b:= b,a+b;
    if ormap(t -> t[2]::odd and t[1] mod 4 = 3, ifactors(b)[2]) then
     R:= R, b; count:= count+1
  fi
od:
R; # Robert Israel, Jan 10 2023

from sympy import factorint
from itertools import islice
def A358311_gen(): # generator of terms
    a, b = 2,1
    while True:
        if any(e&1 and p&3==3 for p, e in factorint(a).items()):
            yield a
        a, b = b, a+b
A358311_list = list(islice(A358311_gen(),40))

phi^n < a(n) < phi^(2n) for n > 4. - Charles R Greathouse IV, Jan 11 2023

A034025 Both primitively and imprimitively represented by x^2+y^2.

Original entry on oeis.org

25, 50, 125, 169, 250, 289, 325, 338, 425, 578, 625, 650, 725, 841, 845, 850, 925, 1025, 1250, 1325, 1369, 1445, 1450, 1525, 1625, 1681, 1682, 1690, 1825, 1850, 2050, 2125, 2197, 2225, 2425, 2525, 2650, 2725, 2738, 2809, 2825, 2873, 2890

Offset: 0

N. J. A. Sloane

nonn

Cf. A001481, A008784, A022544, A034023-.

okQ[n_] := (xy = {x, y} /. {ToRules[Reduce[n == x^2 + y^2, {x, y}, Integers]]}; cnt = Count[xy, {x_, y_} /; GCD[x, y] == 1]; Length[xy] > cnt > 0); Reap[For[n = 1, n <= 3000, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 25 2013 *)

A305553 Numbers that are not the sum of 2 squares and a 4th power.

Original entry on oeis.org

7, 12, 15, 22, 23, 28, 31, 39, 43, 44, 47, 55, 60, 63, 67, 70, 71, 76, 78, 79, 87, 92, 93, 95, 103, 108, 111, 112, 119, 124, 127, 135, 140, 143, 151, 156, 159, 167, 168, 172, 175, 177, 183, 184, 188, 191, 192, 199, 204, 207, 214, 215, 220, 223, 231, 236

Offset: 1

XU Pingya, Jun 20 2018

nonn

Numbers of the form 4*A017101(k) are terms of this sequence.

m is a term iff 16m is also.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.

Subsequence of A000037, A140823 and A022544.
A004215 and A214891 are subsequences.
Cf. A017101, A022552.

n=239;
t=Union@Flatten@Table[x^2+y^2+z^4, {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/4)}];
Complement[Range[0,n], t]

A334736 Dimensions d such that the integer lattice Z^d does not contain the vertices of a regular d-simplex.

Original entry on oeis.org

2, 4, 5, 6, 10, 12, 13, 14, 16, 18, 20, 21, 22, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 50, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98, 100, 101, 102, 104, 106, 108

Offset: 1

Harry Richman, May 08 2020

nonn

List contains d such that (1) d is even and d+1 is not a square, or (2) d == 1 (mod 4) and d+1 is not a sum of two squares; proved by Schoenberg.

			2 is in the list because there is no equilateral triangle in the plane whose vertices all have integer coordinates.
3 is not in the list because there is a regular tetrahedron in space whose vertices have integer coordinates; e.g. (1,1,0), (1,0,1), (0,1,1), (0,0,0).

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.

Complement of A096315.

Cf. A000290, A001481, A022544, A097269.

More terms from Jinyuan Wang, May 09 2020

cp's OEIS Frontend

A071011 Numbers n such that n is a sum of 2 squares (i.e., n is in A001481(k)) and sigma(n) == 0 (mod 4).

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A124981 Odd numbers that are not the sum of 2 squares.

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A173256 Partial sums of A001481.

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A294716 Numbers that are the sum of 2 squares in exactly 5 ways.

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A357018 Records in the number of consecutive integers not expressible as sums of 2 squares.

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A358311 Lucas numbers that are not the sum of two squares.

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Maple

Python

Formula

A034025 Both primitively and imprimitively represented by x^2+y^2.

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Mathematica

A305553 Numbers that are not the sum of 2 squares and a 4th power.

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Author

Keywords

Comments

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Programs

Mathematica

A334736 Dimensions d such that the integer lattice Z^d does not contain the vertices of a regular d-simplex.