A169580 -id:A169580 - cp's OEIS frontend

A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane, Ralph Peterson (ralphp(AT)library.nrl.navy.mil)

All numbers not equal to some 2^k or 5*2^k [Fraser and Gordon]. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006

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

T. D. Noe, Table of n, a(n) for n = 1..1000
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

Complement of A094958. Cf. A169580, A000378, A000419, A000408.

For primitive solutions see A005818.

z=100;lst={};Do[a2=a^2;Do[b2=b^2;Do[c2=c^2;e2=a2+b2+c2;e=Sqrt[e2];If[IntegerQ[e]&&e<=z,AppendTo[lst,e]],{c,b,1,-1}],{b,a,1,-1}],{a,1,z}];Union@lst (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)

is(n)=if(n%5,n,n/5)==2^valuation(n,2) \\ Charles R Greathouse IV, Mar 12 2013

def A005767(n):
    def f(x): return n+x.bit_length()+(x//5).bit_length()
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

a(n) = n + 2*log_2(n) + O(1). - Charles R Greathouse IV, Sep 01 2015

A169580(n) = a(n)^2. - R. J. Mathar, Aug 15 2023

More terms from T. D. Noe, Mar 04 2010

A330893 Numbers whose set of divisors contains a Pythagorean quadruple.

Original entry on oeis.org

42, 72, 84, 126, 144, 156, 168, 198, 210, 216, 252, 288, 294, 312, 330, 336, 342, 360, 378, 396, 420, 432, 462, 468, 504, 546, 570, 576, 588, 594, 624, 630, 648, 660, 672, 684, 714, 720, 756, 780, 792, 798, 840, 864, 882, 900, 924, 930, 936, 966, 990, 1008, 1026

Offset: 1

Michel Lagneau, May 01 2020

nonn

A Pythagorean quadruple (x, y, z, m) is a set of positive integers that satisfy x^2 + y^2 + z^2 = m^2.
The corresponding number of quadruples of the sequence is 1, 1, 2, 2, 2, 1, 3, 1, 3, 2, 4, 3, 2, 2, 1, 4, 1, 2, 3, 2, 7, 4, ... (see the sequence A330894).
It is interesting to note that each set of divisors of a(n) contains m primitive Pythagorean quadruples for some n, m = 1, 2,...
Examples:
- The set of divisors of a(1)= 42 contains only one primitive Pythagorean quadruple: (2, 3, 6, 7).
- The set of divisors of a(9) = 210 contains two primitive Pythagorean quadruples: (2, 3, 6, 7) and (2, 5, 14, 15).
- The set of divisors of a(21) = 420 contains three primitive Pythagorean quadruples: (2, 3, 6, 7), (2, 5, 14, 15) and (4, 5, 20, 21).
If k is in the sequence then so is m*k for m > 1.
Assumes the elements (x,y,z,m) in a quadruple are distinct divisors, as otherwise 6 would be in the sequence with 1^2+2^2+2^2=3^2. - Chai Wah Wu, Nov 16 2020

			168 is in the sequence because the set of divisors  {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168} contains the Pythagorean quadruples {2, 3, 6, 7}, {4, 6, 12, 14} and {8, 12, 24, 28}. The first quadruple is primitive.

Eric Weisstein's World of Mathematics, Pythagorean Quadruples.

Cf. A027750, A169580, A330894, A331365.

with(numtheory):
for n from 3 to 1200 do :
   d:=divisors(n):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^2 + d[j]^2 + d[k]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it>0 then
    printf(`%d, `,n):
    else fi:
   od:

nq[n_] := If[ Mod[n,6]>0, 0, Block[{t, u, v, c = 0, d = Divisors[n], m}, m = Length@ d; Do[ t = d[[i]]^2 + d[[j]]^2; Do[u = t + d[[h]]^2; If[u > n^2, Break[]]; If[ Mod[n^2, u] == 0 && IntegerQ[v = Sqrt@ u] && Mod[n, v] == 0, c++], {h, j+1, m - 1}], {i, m-3}, {j, i+1, m - 2}]; c]]; Select[ Range@ 1026, nq[#] > 0 &] (* Giovanni Resta, May 04 2020 *)

isok(n) = {my(d=divisors(n), x); for (i=1, #d-3, for (j=i+1, #d-2, for (k=j+1, #d-1, if (issquare(d[i]^2 + d[j]^2 + d[k]^2, &x) && !(n % x), return(1)););););} \\ Michel Marcus, Nov 16 2020

a(n) == 0 (mod 6).

A330894 Numbers of Pythagorean quadruples contained in the divisors of A330893(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 3, 1, 3, 2, 4, 3, 2, 2, 1, 4, 1, 2, 3, 2, 7, 4, 2, 2, 8, 2, 1, 4, 4, 2, 3, 7, 3, 2, 5, 2, 2, 4, 6, 2, 5, 2, 11, 6, 4, 1, 4, 1, 6, 2, 4, 12, 2, 5, 1, 4, 6, 4, 2, 5, 6, 4, 1, 2, 3, 4, 17, 6, 2, 3, 6, 1, 5, 6, 1, 3, 4, 6, 6, 13, 1, 2, 4, 8, 4, 4

Offset: 1

Michel Lagneau, May 01 2020

nonn

			a(7) = 3 because A330893(7)=168, and the set of divisors of 168: {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168} contains three Pythagorean quadruples {2, 3, 6, 7}, {4, 6, 12, 14} and {8, 12, 24, 28}.

Cf. A027750, A169580, A330893

with(numtheory):
for n from 3 to 1700 do :
   d:=divisors(n):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^2 + d[j]^2 + d[k]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it>0 then
    printf(`%d, `,it):
    else fi:
   od:

nq[n_] := If[Mod[n, 6] > 0, 0, Block[{t, u, v, c = 0, d = Divisors[n], m}, m = Length@ d; Do[t = d[[i]]^2 + d[[j]]^2; Do[u = t + d[[h]]^2; If[u > n^2, Break[]]; If[Mod[n^2, u] == 0 && IntegerQ[v = Sqrt@ u] && Mod[n, v] == 0, c++], {h, j+1, m-1}], {i, m-3}, {j, i+1, m - 2}]; c]]; Select[Array[nq, 1638], # > 0 &] (* Giovanni Resta, May 04 2020 *)

A331365 Least k whose set of divisors contains exactly n Pythagorean quadruples, or 0 if no such k exists.

Original entry on oeis.org

42, 84, 168, 252, 672, 756, 420, 504, 2592, 1872, 840, 1008, 1512, 2940, 1680, 2016, 1260, 4536, 3360, 3024, 9450, 4620, 5880, 6552, 9504, 6930, 3780, 8400, 23184, 25704, 2520, 6300, 31752, 8820, 19800, 11088, 10920, 13104, 15840, 19152, 19656, 16632, 38016

Offset: 1

Michel Lagneau, May 03 2020

nonn

a(n) == 0 (mod 6).

			a(3) = 168 because the set of the divisors {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168} contains 3 Pythagorean quadruples {2, 3, 6, 7}, {4, 6, 12, 14} and {8, 12, 24, 28}.

Chai Wah Wu, Table of n, a(n) for n = 1..1026
Eric Weisstein's World of Mathematics, Pythagorean Quadruples.

Cf. A027750, A169580, A330893, A330894.

with(numtheory):
for n from 1 to 52 do :
ii:=0:
for q from 3 to 10^8 while(ii=0) do:
   d:=divisors(q):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for l from k+1 to n0 do:
       if d[i]^2 + d[j]^2 + d[k]^2 = d[l]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
   od:
    if it = n
     then
     ii:=1: printf(`%d %d \n`,n,q):
     else
    fi:
od:
od:

upto = 38016; nq[n_] := If[Mod[n, 6] > 0, 0, Block[{t, u, v, c=0, d = Divisors@ n, m}, m = Length@ d; Do[t = d[[i]]^2 + d[[j]]^2; Do[u = t + d[[h]]^2; If[u > n^2, Break[]]; If[Mod[n^2, u] == 0 && IntegerQ[v = Sqrt@ u] && Mod[n, v] == 0, c++], {h, j+1, m-1}], {i, m-3}, {j, i+1, m-2}]; c]]; w = ParallelTable[ {nq@ n, n}, {n, 6 Range[ upto / 6]}]; t=0 Range@ Max[First /@ w]; Do[{q, x} = e; If[q > 0 && t[[q]] == 0, t[[q]] = x], {e, w}]; AppendTo[t, 0]; TakeWhile[t, # > 0 &] (* Giovanni Resta, May 04 2020 *)

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

A217554 Numbers of the form x^2 - 1 that are the sum of 3 nonzero square numbers.

Original entry on oeis.org

3, 24, 35, 48, 99, 120, 168, 195, 224, 288, 323, 360, 440, 483, 528, 675, 728, 840, 899, 1088, 1155, 1224, 1368, 1443, 1680, 1763, 1848, 2024, 2115, 2208, 2400, 2499, 2600, 2808, 2915, 3024, 3248, 3363, 3480, 3720, 3843, 3968, 4224, 4355, 4488, 4760, 4899

Offset: 1

Jon Perry, Oct 06 2012

nonn

Solutions of a^2 - b^2 - c^2 - d^2 = 1 with a, b, c, and d positive.

Integer points on the unit sphere in a Minkowski 4-space.

			3 = 1 + 1 + 1
24 = 16 + 4 + 4
35 = 25 + 9 + 1
48 = 16 + 16 + 16
99 = 81 + 9 + 9
120 = 100 + 16 + 4
168 = 100 + 64 + 4
195 = 169 + 25 + 1
224 = 144 + 64 + 16
288 = 256 + 16 + 16
323 = 289 + 25 + 9
360 = 196 + 100 + 64
440 = 400 + 36 + 4
8 can only be expressed as the sum of 3 squares iff 0 is allowed.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A169580.

nn = 100; t = {}; Do[n = a^2 + b^2 + c^2; If[n <= nn^2 + 2 && IntegerQ[Sqrt[n + 1]], AppendTo[t, n]], {a, nn}, {b, a}, {c, b}]; t = Union[t] (* T. D. Noe, Oct 10 2012 *)

Terms a(14)-a(39) from John W. Layman, Oct 09 2012

A225206 Number of Pythagorean quadruples (a, b, c, d) with d < 10^n.

Original entry on oeis.org

6, 571, 56268, 5614390, 561232920, 56120665334, 5612026652893, 561202243017532, 56120219419339591

Offset: 1

Arkadiusz Wesolowski, May 01 2013

a(n) ~ Pi*A225207(n)/(1+G), where G is Catalan's constant (A006752).

			a(1) = 6 because there are six solutions (a, b, c, d) as follows: (1, 2, 2, 3), (2, 4, 4, 6), (2, 3, 6, 7), (1, 4, 8, 9), (3, 6, 6, 9), (4, 4, 7, 9) with d < 10.

Wikipedia, Pythagorean quadruple

Cf. A169580, A181786, A225207.

a(n) = Sum_{k=1..10^n-1} A181786(k). - Max Alekseyev, Feb 28 2023

a(4) from Giovanni Resta, May 01 2013

a(5)-a(9) from Max Alekseyev, Feb 28 2023

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 *)

A360946 Number of Pythagorean quadruples with inradius n.

Original entry on oeis.org

1, 3, 6, 10, 9, 19, 16, 25, 29, 27, 27, 56, 31, 51, 49, 61, 42, 91, 52, 71, 89, 86, 63, 142, 64, 95, 116, 132, 83, 153, 90, 144, 149, 133, 108, 238, 108, 162, 169, 171, 122, 284, 130, 219, 200, 196, 145, 340, 174, 201, 231, 239, 164, 364, 176, 314, 278, 256, 190, 399, 195, 281, 360, 330

Offset: 1

Miguel-Ángel Pérez García-Ortega, Feb 26 2023

nonn

A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.

For every positive integer n, there is at least one Pythagorean quadruple with inradius n.

			For n=1 the a(1)=1 solution is (1,2,2,3).
For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).
For n=3 the a(3)=6 solutions are (1,6,18,19), (2,5,14,15), (2,6,9,11), (3,4,12,13), (3,6,6,9) and (4,4,7,9).

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.

Miguel-Ángel Pérez García-Ortega, Pythagorean Quadruples (in Spanish).
Wikipedia, Pythagorean quadruple.

Cf. A096907, A096908, A096909, A096910, A097263, A097264, A097265, A097266, A097267

Cf. A169580, A225206, A225207, A230543, A330893, A330894, A331365, A335654, A359741

n=50;
div={};suc={};A={};
Do[A=Join[A,{Range[1,(1+1/Sqrt[3])q]}],{q,1,n}];
Do[suc=Join[suc,{Length[div]}];div={};For [i=1,i<=Length[Extract[A,q]],i++,div=Join[div,Intersection[Divisors[q^2+(Extract[Extract[A,q],i]-q)^2],Range[2(Extract[Extract[A,q],i]-q),Sqrt[q^2+(Extract[Extract[A,q],i]-q)^2]]]]],{q,1,n}];suc=Rest[Join[suc,{Length[div]}]];matriz={{"q"," ","cuaternas"}};For[j=1,j<=n,j++,matriz=Join[matriz,{{j," ",Extract[suc,j]}}]];MatrixForm[Transpose[matriz]]

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A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

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A330893 Numbers whose set of divisors contains a Pythagorean quadruple.

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A330894 Numbers of Pythagorean quadruples contained in the divisors of A330893(n).

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A331365 Least k whose set of divisors contains exactly n Pythagorean quadruples, or 0 if no such k exists.

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

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A217554 Numbers of the form x^2 - 1 that are the sum of 3 nonzero square numbers.

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A225206 Number of Pythagorean quadruples (a, b, c, d) with d < 10^n.

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