A094942 -id:A094942 - cp's OEIS frontend

A000164 Number of partitions of n into 3 squares (allowing part zero).

1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 0, 2, 1, 0, 1, 2, 2, 1, 2, 1, 2, 0, 1, 3, 1, 1, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 3, 0, 1, 2, 1, 2, 0, 2, 2, 0, 1, 3, 3, 1, 2, 2, 1, 0, 2, 2, 3, 2, 1, 2, 1, 0, 1, 4, 2, 2, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 2, 0, 1, 2, 3, 3, 2, 4, 2, 0, 2

Offset: 0

N. J. A. Sloane

nonn

a(n) = number of triples of integers [ i, j, k] such that i >= j >= k >= 0 and n = i^2 + j^2 + k^2. - Michael Somos, Jun 05 2012

			G.f. = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^8 + 2*x^9 + x^10 + x^11 + x^12 + x^13 + ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.

T. D. Noe, Table of n, a(n) for n = 0..10000
Hirschhorn, M. D., Some formulas for partitions into squares, Discrete Math. 211 (2000), pp. 225-228.

Equivalent sequences for other numbers of squares: A000161 (2), A002635 (4), A000174 (5).
Cf. A004215 (positions of zeros), A094942 (positions of ones), A124966 (positions of greater values).
Cf. A005875, A016727.

A000164 := proc(n)
    local a,x,y,z2,z ;
    a := 0 ;
    for x from 0 do
        if 3*x^2 > n then
            return a;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            end if;
            z2 := n-x^2-y^2 ;
            if issqr(z2) then
                z := sqrt(z2) ;
                if z >= y then
                    a := a+1 ;
                end if;
            end if;
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Feb 12 2017

Length[PowersRepresentations[ #, 3, 2]] & /@ Range[0, 104]
e[0,r_,s_,m_]=0;e[n_,r_,s_,m_]:=Length[Select[Divisors[n],Mod[ #,m]==r &]]-Length[Select[Divisors[n],Mod[ #,m]==s &]];alpha[n_]:=5delta[n]+3delta[1/2 n]+4delta[1/3n];beta[n_]:=4e[n,1,3,4]+3e[n,1,7,8]+3e[n,3,5,8];delta[n_]:=If[IntegerQ[Sqrt[n]],1,0];f[n_]:=Table[n-k^2, {k,1,Floor[Sqrt[n]]}]; gamma[n_]:=2 Plus@@(e[ #,1,3,4] &/@f[n]);p3[n_]:=1/12(alpha[n]+beta[n]+gamma[n]);p3[ # ] &/@Range[0,104]
(* Ant King, Oct 15 2010 *)
a[ n_] := If[ n < 0, 0, Sum[ Boole[ n == i^2 + j^2 + k^2], {i, 0, Sqrt[n]}, {j, 0, i}, {k, 0, j}]]; (* Michael Somos, Aug 15 2015 *)

{a(n) = if( n<0, 0, sum( i=0, sqrtint(n), sum( j=0, i, sum( k=0, j, n == i^2 + j^2 + k^2))))}; /* Michael Somos, Jun 05 2012 */

import collections; a = collections.Counter(i*i + j*j + k*k for i in range(100) for j in range(i+1) for k in range(j+1)) # David Radcliffe, Apr 15 2019

Let e(n,r,s,m) be the excess of the number of n's r(mod m) divisors over the number of its s (mod m) divisors, and let delta(n)=1 if n is a perfect square and 0 otherwise. Then, if we define alpha(n) = 5*delta(n) + 3*delta(n/2) + 4*delta(n/3), beta(n) = 4*e(n,1,3,4) + 3*e(n,1,7,8) + 3*e(n,3,5,8), gamma(n) = 2*Sum_{1<=k^2Ant King, Oct 15 2010

Name clarified by Wolfdieter Lang, Apr 08 2013

A124966 Numbers which can be expressed as the ordered sum of 3 squares in 2 or more different ways.

Original entry on oeis.org

9, 17, 18, 25, 26, 27, 29, 33, 34, 36, 38, 41, 45, 49, 50, 51, 53, 54, 57, 59, 61, 62, 65, 66, 68, 69, 72, 73, 74, 75, 77, 81, 82, 83, 85, 86, 89, 90, 94, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 113, 114, 116, 117, 118, 121, 122, 123, 125, 126, 129

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

			a(1)=9 because 9 = 3^2 + 0^2 + 0^2 or 2^2 + 2^2 + 1^2.

Robert Price, Table of n, a(n) for n = 1..83131

Cf. A004215, A094942, A124967-A124971.

Select[Range[129], Length@PowersRepresentations[#, 3, 2] >= 2 &] (* Ray Chandler, Oct 31 2019 *)

Equals = A000027 - A094942 - A004215.

Corrected and extended by Ray Chandler, Nov 30 2006

A224443 Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.

Original entry on oeis.org

41, 50, 54, 65, 66, 74, 86, 90, 98, 99, 110, 113, 114, 117, 121, 122, 126, 131, 137, 145, 150, 164, 166, 169, 174, 178, 179, 181, 182, 186, 197, 200, 205, 216, 218, 219, 222, 226, 227, 229, 237, 258, 260, 264, 265, 275, 286, 291, 296, 302

Offset: 1

Wolfdieter Lang, Apr 08 2013

nonn

These are the numbers for which A000164(a(n)) = 3.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly three ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

			a(1) = 41  = 0^2 + 4^2 + 5^2  = 1^2 + 2^2 + 6^2 = 3^3 + 4^2 + 4^2, and 41 is the first number m with A000164(m) = 3.
The representations [a,b,c] for n = 1, ..., 10, are:
n=1,  41: [0, 4, 5], [1, 2, 6], [3, 4, 4],
n=2,  50: [0, 1, 7], [0, 5, 5], [3, 4, 5],
n=3,  54: [1, 2, 7], [2, 5, 5], [3, 3, 6],
n=4,  65: [0, 1, 8], [0, 4, 7], [2, 5, 6],
n=5,  66: [1, 1, 8], [1, 4, 7], [4, 5, 5],
n=6,  74: [0, 5, 7], [1, 3, 8], [3, 4, 7],
n=7,  86: [1, 2, 9], [1, 6, 7], [5, 5, 6],
n=8,  90: [0, 3, 9], [1, 5, 8], [4, 5, 7],
n=9,  98: [0, 7, 7], [1, 4, 9], [3, 5, 8],
n=10, 99: [1, 7, 7], [3, 3, 9], [5, 5, 7].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000164, A005875, A000378, A094942 (one way), A224442 (two ways).

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^23, 4, min(4, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
    end:
a:= proc(n) option remember; local k;
      for k from 1 +`if`(n=1, 0, a(n-1))
      while b(k, isqrt(k), 3)<>3 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 09 2013

Select[ Range[0, 400], Length[ PowersRepresentations[#, 3, 2]] == 3 &] (* Jean-François Alcover, Apr 09 2013 *)

This sequence gives the increasingly ordered numbers of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly three such representations}.

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 3, m >= 0}.

A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

Original entry on oeis.org

9, 17, 18, 25, 26, 27, 29, 33, 34, 36, 38, 45, 49, 51, 53, 57, 59, 61, 62, 68, 69, 72, 73, 75, 77, 82, 83, 85, 94, 97, 100, 102, 104, 105, 106, 107, 108, 109, 116, 118, 123, 130, 132, 136, 138, 139, 141, 144, 147, 152, 154, 155, 157, 158, 165, 177, 180, 187

Offset: 1

Wolfdieter Lang, Apr 08 2013

nonn

These are the numbers for which A000164(a(n)) = 2.
a(n) is the n-th largest number which has a representation as sum of three integer squares (square 0 allowed), in exactly two ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity for a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

			a(1) = 9 = 0^2 + 0^2 + 3^2 = 1^2 + 2^2 + 2^2, and 9 is the smallest number m with A000164(m) = 2 for m >= 0. The multiplicity with order and signs taken into account is 2*3 + 8*3 = 30 = A005875(9).
The two representations [a,b,c] for a(n), n = 1, ..., 10, are
n=1,   9 = [0, 0, 3], [1, 2, 2],
n=2,  17 = [0, 1, 4], [2, 2, 3],
n=3,  18 = [0, 3, 3], [1, 1, 4],
n=4,  25 = [0, 0, 5], [0, 3, 4],
n=5,  26 = [0, 1, 5], [1, 3, 4],
n=6,  27 = [1, 1, 5], [3, 3, 3],
n=7,  29 = [0, 2, 5], [2, 3, 4],
n=8,  33 = [1, 4, 4], [2, 2, 5],
n=9,  34 = [0, 3, 5], [3, 3, 4],
n=10, 36 = [0, 0, 6], [2, 4, 4].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000164, A005875, A000378, A094942 (one way), A224443 (three ways).

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^22, 3, min(3, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
    end:
a:= proc(n) option remember; local k;
      for k from 1 +`if`(n=1, 0, a(n-1))
      while b(k, isqrt(k), 3)<>2 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 09 2013

Select[ Range[0, 200], Length[ PowersRepresentations[#, 3, 2]] == 2 &] (* Jean-François Alcover, Apr 09 2013 *)

This sequence gives the increasingly ordered elements of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly two such representation}.

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 2, m >= 0}.

A294577 Numbers that are the sum of three squares (square 0 allowed) in exactly four ways.

Original entry on oeis.org

81, 89, 101, 125, 129, 134, 149, 161, 162, 170, 171, 173, 189, 198, 201, 233, 241, 242, 243, 245, 246, 249, 250, 251, 254, 270, 274, 278, 285, 289, 294, 299, 324, 339, 349, 356, 361, 363, 370, 371, 378, 387, 390, 393, 395, 404, 406, 411, 417, 429, 433, 451

Offset: 1

Robert Price, Nov 02 2017

nonn

These are the numbers for which A000164(a(n)) = 4.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly four ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..1030

Cf. A000164, A005875, A000378, A094942, A224442, A224443.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 4 &]

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

A294594 Numbers that are the sum of three squares (square 0 allowed) in exactly five ways.

Original entry on oeis.org

146, 153, 185, 206, 221, 225, 230, 234, 257, 261, 266, 293, 305, 325, 338, 350, 353, 354, 362, 377, 381, 398, 402, 405, 409, 410, 413, 414, 419, 437, 470, 474, 477, 481, 491, 514, 525, 539, 557, 563, 579, 582, 584, 586, 590, 611, 612, 625, 630, 635, 638, 642

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 5.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly five ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..843

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 5 &]

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

A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

Original entry on oeis.org

194, 209, 269, 281, 290, 297, 321, 326, 329, 342, 365, 386, 389, 401, 426, 434, 449, 459, 482, 485, 489, 497, 513, 531, 534, 542, 546, 554, 558, 561, 578, 601, 602, 633, 649, 659, 662, 665, 675, 678, 681, 693, 699, 705, 713, 714, 722, 737, 741, 747, 750, 754

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 6.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly six ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..1400

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 6 &]

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

A124978 Smallest positive number which has exactly n different partitions as a sum of 4 squares x^2+y^2+z^2+t^2.

Original entry on oeis.org

1, 4, 18, 34, 50, 66, 82, 114, 90, 130, 150, 178, 162, 198, 318, 210, 250, 234, 322, 406, 465, 330, 306, 402, 462, 390, 474, 378, 490, 486, 654, 610, 522, 450, 778, 678, 642, 570, 666, 726, 594, 714, 770, 774, 986, 630, 738, 945, 1035, 850, 1222, 978, 1014, 918

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

Is it known that a(n) always exists? - Franklin T. Adams-Watters, Dec 18 2006

A002635(a(n)) = n. - Reinhard Zumkeller, Jul 13 2014

			a(4)=34 because 34 is smallest number which has 4 partitions 34=4^2+3^2+3^2+0^2 = 4^2+4^2+1^2+1^2 = 5^2+2^2+2^2+1^2 = 5^2+3^2+0^2+0^2
a(3)=18 which has 3 partitions 18=0^2+0^2+3^2+3^2=0^2+1^2+1^2+4^2=1^2+2^2+2^2+3^2.

Cf. A006431, A094942, A124979-A124983, A000378, A002635, A061262.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a124978 = (+ 1) . fromJust . (`elemIndex` (tail a002635_list))
-- Reinhard Zumkeller, Jul 13 2014

kmin[n_] := If[n<5, 1, 10n](* empirical, should be lowered in case of doubt *);
a[n_] := a[n] = For[k=kmin[n], True, k++, If[Length[PowersRepresentations[ k, 4, 2]] == n, Return[k]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 1000}] (* Jean-François Alcover, Mar 11 2019 *)

cnt4sqr(n)={ local(cnt=0,t2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), for(z=y,floor(n-x^2-y^2), t2=n-x^2-y^2-z^2 ; if( t2>=z^2 && issquare(n-x^2-y^2-z^2), cnt++ ; ) ; ) ; ) ; ) ; return(cnt) ; }
A124978(n)= { local(a=1) ; while(1, if( cnt4sqr(a)==n, return(a) ; ) ; a++ ; ) ; }
{ for(n=1,100, print(n," ",A124978(n)) ; ) ; } \\ R. J. Mathar, Nov 29 2006

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

More terms from Franklin T. Adams-Watters, Dec 18 2006

A294710 Numbers that are the sum of three squares (square 0 allowed) in exactly seven ways.

Original entry on oeis.org

306, 314, 341, 441, 450, 458, 494, 506, 581, 585, 593, 605, 654, 657, 674, 698, 706, 726, 731, 738, 746, 773, 806, 842, 850, 873, 890, 891, 893, 894, 899, 901, 905, 906, 934, 978, 985, 998, 1011, 1013, 1019, 1050, 1058, 1061, 1067, 1073, 1086, 1094, 1101

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 7.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly seven ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..988

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594, A294595.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 7 &]

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

A294711 Numbers that are the sum of three squares (square 0 allowed) in exactly eight ways.

Original entry on oeis.org

369, 374, 446, 461, 486, 509, 530, 549, 566, 621, 641, 666, 677, 686, 710, 749, 770, 789, 797, 818, 821, 825, 833, 849, 869, 882, 902, 945, 954, 962, 969, 971, 981, 1010, 1014, 1017, 1022, 1029, 1069, 1085, 1098, 1146, 1157, 1174, 1221, 1233, 1242, 1245

Offset: 1

Robert Price, Nov 07 2017

nonn

These are the numbers for which A000164(a(n)) = 8.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly eight ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

Robert Price, Table of n, a(n) for n = 1..1701

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594, A294595, A294710.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 8 &]

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

cp's OEIS Frontend

A000164 Number of partitions of n into 3 squares (allowing part zero).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A124966 Numbers which can be expressed as the ordered sum of 3 squares in 2 or more different ways.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A224443 Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A294577 Numbers that are the sum of three squares (square 0 allowed) in exactly four ways.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A294594 Numbers that are the sum of three squares (square 0 allowed) in exactly five ways.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs