A124966 -id:A124966 - 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

A124967 Numbers which can be expressed as the ordered sum of 3 squares in 3 or more different ways.

Original entry on oeis.org

41, 50, 54, 65, 66, 74, 81, 86, 89, 90, 98, 99, 101, 110, 113, 114, 117, 121, 122, 125, 126, 129, 131, 134, 137, 145, 146, 149, 150, 153, 161, 162, 164, 166, 169, 170, 171, 173, 174, 178, 179, 181, 182, 185, 186, 189, 194, 197, 198, 200, 201, 205, 206, 209

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

			a(1) = 41 because 41 = 4^2+4^2+3^2 or 5^2+4^2+0^2 or 6^2+2^2+1^2.
117=0^2+6^2+9^2=1^2+4^2+10^2=2^2+7^2+8^2, so 117 is in the list.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A124966-A124971.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^23, 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)<3 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 10 2013

Select[Range[210], Length@PowersRepresentations[#, 3, 2] >= 3 &] (* Ray Chandler, Oct 31 2019 *)

isA124967(n)={ local(cnt=0,z2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), z2=n-x^2-y^2 ; if( z2>=y^2 && issquare(z2), cnt++ ; ) ; if(cnt >=3, return(1) ) ; ) ; ) ; return(0) ; } { for(n=1,200, if( isA124967(n), print1(n,", ") ; ) ; ) ; } (Mathar)

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

A124968 Numbers which can be expressed as the ordered sum of 3 squares in 4 or more different ways.

Original entry on oeis.org

81, 89, 101, 125, 129, 134, 146, 149, 153, 161, 162, 170, 171, 173, 185, 189, 194, 198, 201, 206, 209, 221, 225, 230, 233, 234, 241, 242, 243, 245, 246, 249, 250, 251, 254, 257, 261, 266, 269, 270, 274, 278, 281, 285, 289, 290, 293, 294, 297, 299, 305, 306

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

Subset of A124967.

			a(1)=81 because 81 = 6^2 + 6^2 + 3^2 = 7^2 + 4^2 + 4^2 = 8^2 + 4^2 + 1^2 = 9^2 + 0^2 + 0^2.
161 = 1^2 + 4^2 + 12^2 = 2^2 + 6^2 + 11^2 = 4^2 + 8^2 + 9^2 = 5^2 + 6^2 + 10^2, so 161 is in the list.

Robert Price, Table of n, a(n) for n = 1..81953

Cf. A124966-A124971.

Select[Range[310], Length@PowersRepresentations[#, 3, 2] >= 4 &] (* Ray Chandler, Oct 31 2019 *)

isA124968(n)={ local(cnt=0,z2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), z2=n-x^2-y^2 ; if( z2>=y^2 && issquare(z2), cnt++ ; ) ; if(cnt >=4, return(1) ) ; ) ; ) ; return(0) ; } { for(n=1,800, if( isA124968(n), print1(n,", ") ; ) ; ) ; } \\ R. J. Mathar

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

A124971 Numbers n which can be expressed as the ordered sum of 3 squares in 2 or more different ways and such that n+1 has the same property.

Original entry on oeis.org

17, 25, 26, 33, 49, 50, 53, 61, 65, 68, 72, 73, 74, 81, 82, 85, 89, 97, 98, 99, 100, 101, 104, 105, 106, 107, 108, 109, 113, 116, 117, 121, 122, 125, 129, 130, 131, 136, 137, 138, 144, 145, 146, 149, 152, 153, 154, 157, 161, 164, 165, 169, 170, 173, 177, 178

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

			a(1)=17 because 17=3^2+2^2+2^2 = 4^2+1^2+0^2 and a(1)+1= 18=3^2+3^2+0^2 = 4^2+1^2+1^2

R. J. Mathar, Table of n, a(n) for n = 1..6598

Cf. A124966-A124970.

Select[Range[200], Length@PowersRepresentations[#, 3, 2] > 1 && Length@PowersRepresentations[# + 1, 3, 2] > 1 &] (* Ray Chandler, Oct 31 2019 *)

isCnt3sqr(n)={ local(cnt=0,z2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), z2=n-x^2-y^2 ; if( z2>=y^2 && issquare(z2), cnt++ ; ) ; if(cnt >=2, return(1) ) ; ) ; ) ; return(0) ; } isA124971(n)= { return( isCnt3sqr(n) && isCnt3sqr(n+1)) ; } { for(n=1,200, if( isA124971(n), print1(n,", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 29 2006

A000164(n)>=2 and A000164(n+1)>=2. - R. J. Mathar, Nov 29 2006

Corrected and extended by Ray Chandler, Nov 30 2006

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

A124970 Smallest positive integer which can be expressed as the ordered sum of 3 squares in exactly n different ways.

Original entry on oeis.org

7, 1, 9, 41, 81, 146, 194, 306, 369, 425, 594, 689, 866, 1109, 1161, 1154, 1361, 1634, 1781, 1889, 2141, 2729, 2609, 3626, 3366, 3566, 3449, 3506, 4241, 4289, 4826, 5066, 5381, 7034, 5561, 6254, 7229, 7829, 8186, 8069, 8126, 8609, 8921, 8774, 10386, 11574, 11129

Offset: 0

Artur Jasinski, Nov 14 2006, Nov 20 2006

nonn

Michael S. Branicky, Table of n, a(n) for n = 0..2102

Cf. A000451, A095809, A000437, A122699, A124966-A124971.

f[n_] := Block[{k = 1}, While[Length@PowersRepresentations[k, 3, 2] != n, k++]; k]; Table[f[n], {n, 0, 44}] (* Ray Chandler, Oct 31 2019 *)

from collections import Counter
from itertools import count, combinations_with_replacement as mc
def aupto(lim):
  sq = filter(lambda x: x<=lim, (i**2 for i in range(int(lim**(1/2))+2)))
  s3 = filter(lambda x: 0Michael S. Branicky, Jul 01 2021

Extended by Ray Chandler, Nov 30 2006

a(45) and beyond from Michael S. Branicky, Jul 01 2021

A124969 Numbers which can be expressed as an ordered sum of 3 squares in 5 or more different ways.

Original entry on oeis.org

146, 153, 185, 194, 206, 209, 221, 225, 230, 234, 257, 261, 266, 269, 281, 290, 293, 297, 305, 306, 314, 321, 325, 326, 329, 338, 341, 342, 350, 353, 354, 362, 365, 369, 374, 377, 381, 386, 389, 398, 401, 402, 405, 409, 410, 413, 414, 419, 425, 426, 434

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

			a(1)=146 because 146=9^2+7^2+4^2 = 9^2+8^2+1^2 = 11^2+4^2+3^2 = 11^2+5^2+0^2 = 12^2+1^2+1^2
185=0^2+4^2+13^2=0^2+8^2+11^2=2^2+9^2+10^2=4^2+5^2+12^2=6^2+7^2+10^2, so 185 is in the list.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A124966-A124971.

Select[Range[434], Length@PowersRepresentations[#, 3, 2] >= 5 &] (* Ray Chandler, Oct 31 2019 *)

isA124969(n)={ local(cnt=0,z2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), z2=n-x^2-y^2 ; if( z2>=y^2 && issquare(z2), cnt++ ; ) ; if(cnt >=5, return(1) ) ; ) ; ) ; return(0) ; } { for(n=1,800, if( isA124969(n), print1(n,", ") ; ) ; ) ; } \\ R. J. Mathar, Dec 07 2006

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

A294578 Numbers which can be expressed as an ordered sum of 3 squares in 6 or more different ways.

Original entry on oeis.org

194, 209, 269, 281, 290, 297, 306, 314, 321, 326, 329, 341, 342, 365, 369, 374, 386, 389, 401, 425, 426, 434, 441, 446, 449, 450, 458, 459, 461, 482, 485, 486, 489, 494, 497, 506, 509, 513, 521, 530, 531, 534, 542, 545, 546, 549, 554, 558, 561, 566, 569, 578

Offset: 1

Robert Price, Nov 02 2017

nonn

Robert Price, Table of n, a(n) for n = 1..80080

Cf. A000164, A094942, A124966, A124967, A124968.

Select[Range[578], Length[PowersRepresentations[#, 3, 2]] >= 6 &]

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

A294596 Numbers which can be expressed as an ordered sum of 3 squares in 7 or more different ways.

Original entry on oeis.org

306, 314, 341, 369, 374, 425, 441, 446, 450, 458, 461, 486, 494, 506, 509, 521, 530, 545, 549, 566, 569, 581, 585, 593, 594, 605, 614, 621, 626, 629, 641, 650, 654, 657, 666, 674, 677, 686, 689, 698, 701, 706, 710, 725, 726, 729, 731, 734, 738, 746, 749, 761

Offset: 1

Robert Price, Nov 03 2017

nonn

Robert Price, Table of n, a(n) for n = 1..78680

Cf. A000164, A094942, A124966, A124967, A124968, A124969, A294578.

Select[Range[761], Length[PowersRepresentations[#, 3, 2]] >= 7 &]

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

A294597 Numbers which can be expressed as an ordered sum of 3 squares in 8 or more different ways.

Original entry on oeis.org

369, 374, 425, 446, 461, 486, 509, 521, 530, 545, 549, 566, 569, 594, 614, 621, 626, 629, 641, 650, 666, 677, 686, 689, 701, 710, 725, 729, 734, 749, 761, 770, 774, 789, 794, 797, 801, 809, 810, 818, 821, 825, 833, 845, 846, 849, 854, 857, 866, 869, 881, 882

Offset: 1

Robert Price, Nov 03 2017

nonn

Robert Price, Table of n, a(n) for n = 1..77692

Cf. A000164, A094942, A124966, A124967, A124968.

Select[Range[1000],Length[PowersRepresentations[#,3,2]]>7&] (* Harvey P. Dale, Jul 03 2019 *)

A294714 Numbers which can be expressed as an ordered sum of 3 squares in 9 or more different ways.

Original entry on oeis.org

425, 521, 545, 569, 594, 614, 626, 629, 650, 689, 701, 725, 729, 734, 761, 774, 794, 801, 809, 810, 845, 846, 854, 857, 866, 881, 909, 914, 926, 929, 941, 950, 953, 965, 974, 986, 989, 990, 1001, 1025, 1026, 1034, 1041, 1046, 1049, 1053, 1062, 1070, 1074

Offset: 1

Robert Price, Nov 07 2017

nonn

Robert Price, Table of n, a(n) for n = 1..75991

Cf. A000164, A094942, A124966, A124967, A124968.

Select[Range[882], Length[PowersRepresentations[#, 3, 2]] >= 9 &]

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).

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A124967 Numbers which can be expressed as the ordered sum of 3 squares in 3 or more different ways.

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A124968 Numbers which can be expressed as the ordered sum of 3 squares in 4 or more different ways.

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A124971 Numbers n which can be expressed as the ordered sum of 3 squares in 2 or more different ways and such that n+1 has the same property.

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A124970 Smallest positive integer which can be expressed as the ordered sum of 3 squares in exactly n different ways.

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A124969 Numbers which can be expressed as an ordered sum of 3 squares in 5 or more different ways.

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Mathematica

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A294578 Numbers which can be expressed as an ordered sum of 3 squares in 6 or more different ways.

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