A002635 -id:A002635 - cp's OEIS frontend

A000161 Number of partitions of n into 2 squares.

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter.
Number of similar sublattices of square lattice with index n.
Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013
a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011

			25 = 3^2+4^2 = 5^2, so a(25) = 2.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339

T. D. Noe, Table of n, a(n) for n = 0..10000
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
Henry Bottomley, Illustration of initial terms
R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Michael Gilleland, Some Self-Similar Integer Sequences
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 84.
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228. [From _Ant King_, Oct 05 2010]
Index entries for sequences related to sublattices
Index entries for sequences related to sums of squares
Index entries for "core" sequences

Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290.
Cf. A000925, A247367.
Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5).

a000161 n =
   sum $ map (a010052 . (n -)) $ takeWhile (<= n `div` 2) a000290_list
a000161_list = map a000161 [0..]
-- Reinhard Zumkeller, Aug 16 2011

A000161 := proc(n) local i,j,ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ];
A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007

Length[PowersRepresentations[ #,2,2]] &/@Range[0,150] (* Ant King, Oct 05 2010 *)

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))) \\ for illustrative purpose

A000161(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n),issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014, improves earlier code by M. F. Hasler, Nov 23 2007

A000161(n)=#sum2sqr(n) \\ See A133388 for sum2sqr(). - M. F. Hasler, May 13 2018

from math import prod
from sympy import factorint
def A000161(n):
    f = factorint(n)
    return int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 1 # Chai Wah Wu, Sep 08 2022

a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007

Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010

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

Original entry on oeis.org

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

A006431 Numbers that have a unique partition into a sum of four nonnegative squares.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224, 384, 512, 896, 1536, 2048, 3584, 6144, 8192, 14336, 24576, 32768, 57344, 98304, 131072, 229376, 393216, 524288, 917504, 1572864, 2097152, 3670016, 6291456, 8388608, 14680064

Offset: 1

David M. Bloom

From a(16) = 96 onwards, the terms of this sequence satisfy the third-order recurrence relation a(n) = 4a(n-3). - Ant King, Aug 15 2010

A002635(a(n)) = 1. - Reinhard Zumkeller, Jul 13 2014

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (0,0,4).

Cf. A002635, A180149, A245022.

a006431 n = a006431_list !! (n-1)
a006431_list = filter ((== 1) . a002635) [0..]
-- Reinhard Zumkeller, Jul 13 2014

Select[Range[0,3584], Length[PowersRepresentations[ #,4,2]] == 1&] (* Ant King, Aug 15 2010 *)
CoefficientList[Series[x  (36 x^13 + 28 x^12 + 32 x^11 + 21 x^10 + 17 x^9 + 14 x^8 + 13 x^7 + 12 x^6 + 5 x^5 + 2 x^4 - x^3 - 3 x^2 - 2 x - 1)/(4 x^3 - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{0,0,4},{0,1,2,3,5,6,7,8,11,14,15,23,24,32,56},50] (* Harvey P. Dale, Nov 26 2015 *)

{a(n)=if(n<2, 0, if(n<15, [1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32] [n-1], [4, 7, 12][n%3+1]*2^(n\3*2-7)))} /* Michael Somos, Apr 23 2006 */

Consists of the seven odd numbers 1, 3, 5, 7, 11, 15, 23, plus 0, and numbers of forms 2*4^k, 6*4^k, 14*4^k, k >= 0.
The set {n nonnegative : A002635(n) = 1}.
G.f.: x^2*(36*x^13 +28*x^12 +32*x^11 +21*x^10 +17*x^9 +14*x^8 +13*x^7 +12*x^6 +5*x^5 +2*x^4 -x^3 -3*x^2 -2*x -1) / (4*x^3 -1). - Colin Barker, Apr 20 2013
log(a(n)) = n*log(4)/3 + C(n) + o(1) where C(n) ~ {-2.82922, -3.00364, -2.90612} for n (mod 3) == {2,0,1}. - Bill McEachen, Oct 21 2022

More terms from James Sellers, Dec 24 1999
Corrected by T. D. Noe, Jun 15 2006
Definition revised by Ant King, May 06 2010
Edited and Grosswald reference added by Wolfdieter Lang, Aug 12 2015

A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 4, 3, 2, 2, 2, 2, 3, 3, 1, 4, 4, 2, 2, 3, 2, 3, 4, 2, 3, 3, 2, 4, 3, 2, 4, 4, 2, 4, 4, 1, 4, 5, 1, 2, 3, 4, 6, 4, 3, 2, 5, 2, 3, 3, 3, 6, 5, 2, 2, 5, 3, 5, 4, 2, 4, 5, 3, 4, 5, 2, 4, 6, 2, 6, 3, 3, 6, 3, 2, 3, 7, 3, 6, 6, 2, 4, 6, 3, 2

Offset: 0

N. J. A. Sloane, Sep 18 2001

Fermat asserted that every number is the sum of three triangular numbers. This was proved by Gauss, who recorded in his Tagebuch entry for Jul 10 1796 that: EYPHKA! num = DELTA + DELTA + DELTA.
a(n) <= A167618(n). - Reinhard Zumkeller, Nov 07 2009
Equivalently, number of ways of writing n as an unordered sum of exactly 3 triangular numbers. - Jon E. Schoenfield, Mar 28 2021

			0 : empty sum
1 : 1
2 : 1+1
3 : 3 = 1+1+1
4 : 3+1
5 : 3+1+1
6 : 6 = 3+3
7 : 6+1 = 3+3+1
...
13 : 10 + 3 = 6 + 6 + 1, so a(13) = 2.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. (8).
D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302.
G. Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
Mel Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian). Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Cf. A007294, A053604, A008443, A063993, A061262.

# reuses code in A000217
A002636 := proc(n)
    local a,i,Ti, j,Tj, Tk ;
    a := 0 ;
    for i from 0 do
        Ti := A000217(i) ;
        if Ti > n then
            break ;
        end if;
        for j from i do
            Tj := A000217(j) ;
            if Ti+Tj > n then
                break ;
            end if;
            Tk := n-Ti-Tj ;
            if Tk >= Tj and isA000217(Tk) then
                a := a+1 ;
            end if;
            if Tk < Tj then
                break ;
            end if;
        end do:
    end do:
    a ;
end proc:
seq(A002636(n),n=0..40) ; # R. J. Mathar, May 26 2025

a = Table[ n(n + 1)/2, {n, 0, 15} ]; b = {0}; c = Table[ 0, {100} ]; Do[ b = Append[ b, a[ [ i ] ] + a[ [ j ] ] + a[ [ k ] ] ], {k, 1, 15}, {j, 1, k}, {i, 1, j} ]; b = Delete[ b, 1 ]; b = Sort[ b ]; l = Length[ b ]; Do[ If[ b[ [ n ] ] < 100, c[ [ b[ [ n ] ] + 1 ] ]++ ], {n, 1, l} ]; c

first(n)=my(v=vector(n+1),A,B,C); for(a=0,n, A=a*(a+1)/2; if(A>n, break); for(b=0,a, B=A+b*(b+1)/2; if(B>n, break); for(c=0,b, C=B+c*(c+1)/2; if(C>n, break); v[C+1]++))); v \\ Charles R Greathouse IV, Jun 23 2017

More terms from Robert G. Wilson v, Sep 20 2001

Entry revised by N. J. A. Sloane, Feb 25 2007

A000174 Number of partitions of n into 5 squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 3, 3, 3, 4, 3, 3, 2, 2, 4, 4, 4, 4, 5, 3, 3, 4, 3, 5, 5, 5, 6, 5, 3, 5, 5, 5, 6, 5, 6, 5, 4, 4, 6, 7, 6, 8, 8, 7, 5, 6, 5, 8, 8, 4, 9, 7, 6, 7, 7, 8, 9, 9, 7, 9, 6, 7, 8, 10, 9, 9, 11, 7, 8, 8, 8, 11, 11, 9, 12, 11, 7, 9, 9, 12, 13, 9, 10, 11, 8, 6, 10, 13, 13, 13

Offset: 0

N. J. A. Sloane

nonn

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

Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A002635 (4), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).

Cf. A025429, A295160 (largest number k with a(k) = n).

Table[PowersRepresentations[n, 5, 2] // Length, {n, 0, 100}] (* Jean-François Alcover, Feb 04 2016 *)

A180149 Integers with precisely two partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

4, 9, 10, 12, 13, 16, 17, 19, 20, 21, 22, 29, 30, 31, 35, 39, 40, 44, 46, 47, 48, 64, 71, 80, 88, 120, 160, 176, 184, 192, 256, 320, 352, 480, 640, 704, 736, 768, 1024, 1280, 1408, 1920, 2560, 2816, 2944, 3072, 4096, 5120, 5632, 7680

Offset: 1

Ant King, Aug 17 2010

The largest odd member of this sequence is 71, and from a(32)=320 onwards the terms satisfy the eighth-order recurrence relation a(n)=4a(n-8).

A002635(a(n)) = 2. - Reinhard Zumkeller, Jul 13 2014

			As the fifth integer which has precisely two partitions into sums of four squares of nonnegative numbers is 13, then a(5)=13.

Robert Price, Table of n, a(n) for n = 1..65
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A002635, A006431, A245022.

a180149 n = a180149_list !! (n-1)
a180149_list = filter ((== 2) . a002635) [0..]
-- Reinhard Zumkeller, Jul 13 2014

Select[Range[10000], Length[PowersRepresentations[ #, 4, 2]]==2&]

The members of this sequence are {9, 13, 17, 19, 21, 29, 30, 31, 35, 39, 46, 47, 71} together with all integers of the form 5*2^N, 11*2^N and {1,3,30,46}*4^N where N > 0 (which includes a necessary correction to Lehmer's result).

A245022 Integers with precisely three partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

18, 25, 26, 27, 28, 33, 37, 38, 41, 43, 51, 53, 55, 59, 60, 62, 72, 79, 92, 95, 104, 112, 152, 240, 248, 288, 368, 416, 448, 608, 960, 992, 1152, 1472, 1664, 1792, 2432, 3840, 3968, 4608, 5888, 6656, 7168, 9728, 15360, 15872, 18432, 23552, 26624, 28672

Offset: 1

Reinhard Zumkeller, Jul 13 2014

nonn

A002635(a(n)) = 3.

			a(1) = 18 = 16 + 1 + 1 + 0 = 9 + 9 + 0 + 0 = 9 + 4 + 4 + 1;
a(2) = 25 = 25 + 0 + 0 + 0 = 16 + 9 + 0 + 0 = 16 + 4 + 4 + 1;
a(3) = 26 = 25 + 1 + 0 + 0 = 16 + 9 + 1 + 0 = 9 + 9 + 4 + 4;
a(4) = 27 = 25 + 1 + 1 + 0 = 16 + 9 + 1 + 1 = 9 + 9 + 9 + 0;
a(5) = 28 = 25 + 1 + 1 + 1 = 16 + 4 + 4 + 4 = 9 + 9 + 9 + 1;
a(6) = 33 = 25 + 4 + 4 + 0 = 16 + 16 + 1 + 0 = 16 + 9 + 4 + 4;
a(7) = 37 = 36 + 1 + 0 + 0 = 25 + 4 + 4 + 4 = 16 + 16 + 4 + 1;
a(8) = 38 = 36 + 1 + 1 + 0 = 25 + 9 + 4 + 0 = 16 + 9 + 9 + 4;
a(9) = 41 = 36 + 4 + 1 + 0 = 25 + 16 + 0 + 0 = 16 + 16 + 9 + 0;
a(10) = 43 = 25 + 16 + 1 + 1 = 25 + 9 + 9 + 0 = 16 + 9 + 9 + 9;
a(11) = 51 = 49 + 1 + 1 + 0 = 25 + 25 + 1 + 0 = 25 + 16 + 9 + 1;
a(12) = 53 = 49 + 4 + 0 + 0 = 36 + 16 + 1 + 0 = 36 + 9 + 4 + 4.

Robert Price, Table of n, a(n) for n = 1..55
Index entries for sequences related to sums of squares

Cf. A002635, A006431, A180149.

a245022 n = a245022_list !! (n-1)
a245022_list = filter ((== 3) . a002635) [0..]

Select[ Range@ 30000, Length@PowersRepresentations[#, 4, 2] == 3 &] (* Robert G. Wilson v, Oct 27 2017 *)

a(44)-a(50) from Robert Price, Oct 26 2017

A294282 Integers with precisely four partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

34, 36, 42, 45, 49, 57, 61, 63, 65, 67, 68, 69, 77, 78, 83, 87, 94, 107, 116, 119, 136, 144, 168, 272, 312, 376, 464, 544, 576, 672, 1088, 1248, 1504, 1856, 2176, 2304, 2688, 4352, 4992, 6016, 7424, 8704, 9216, 10752, 17408, 19968, 24064, 29696, 34816, 36864, 43008, 69632, 79872, 96256

Offset: 1

Robert Price, Oct 26 2017

nonn

A002635(a(n)) = 4.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A002635, A006431, A180149, A245022.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==4 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294297 Integers with precisely five partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

50, 52, 54, 58, 70, 73, 74, 75, 76, 84, 85, 86, 89, 91, 93, 101, 103, 109, 111, 113, 127, 131, 140, 142, 143, 151, 167, 191, 200, 208, 216, 232, 280, 296, 304, 336, 344, 560, 568, 800, 832, 864, 928, 1120, 1184, 1216, 1344, 1376, 2240, 2272, 3200, 3328, 3456

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 5.

Robert Price, Table of n, a(n) for n = 1..81
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282.

f[n_] := Length@ PowersRepresentations[n, 4, 2]; Select[ Range@ 3500, f@# == 5 &] (* Robert G. Wilson v, Oct 27 2017 *)

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

cp's OEIS Frontend

A000161 Number of partitions of n into 2 squares.

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

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A006431 Numbers that have a unique partition into a sum of four nonnegative squares.

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A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

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A000174 Number of partitions of n into 5 squares.

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A180149 Integers with precisely two partitions into sums of four squares of nonnegative numbers.

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