A025428 -id:A025428 - cp's OEIS frontend

A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Also parity of the divisor function A000005 if n >= 1. - Omar E. Pol, Jan 14 2012
This sequence can be considered as k=1 analog of A025426 (k=2), A025427 (k=3), A025428 (k=4); see also A000161. - M. F. Hasler, Jan 25 2013
Also, the decimal expansion of Sum_{n >= 0} 1/(10^n)^n. -  Eric Desbiaux, Mar 15 2009, rephrased and simplified by M. F. Hasler, Jan 26 2013
Run lengths of zeros gives A005843, the nonnegative even numbers. - Jeremy Gardiner, Jan 14 2018
Inverse Möbius transform of Liouville's lambda function (A008836), n >= 1. - Wesley Ivan Hurt, Jun 22 2024

			G.f. = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 + x^81 + ...

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 3-4, also p. 166, Ex. 5.5.1.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, Problem 20.
Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (11.14).
Michael D. Hirschhorn, The Power of q, Springer, 2017. See phi(q) page 8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 55.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
David Christopher and Tamil Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), Article 15.11.5.
Robert Price, Comments on A010052 concerning Elementary Cellular Automata, Jan 29 2016.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Stephen Wolfram, A New Kind of Science.
Index entries for characteristic functions
Index to Elementary Cellular Automata.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to cellular automata.

Column k=1 of A243148, A337165, A341040 (for n>0).
Cf. A000005, A000122, A005369, A007913, A008836 (Mobius transf.), A037011, A063524, A258998, A271102 (Dirichlet inv), A046951 (inv. Mobius trans.).
First differences of A000196.

a010052 n = fromEnum $ a000196 n ^ 2 == n
-- Reinhard Zumkeller, Jan 26 2012, Feb 20 2011
a010052_list = concat (iterate (\xs -> xs ++ [0,0]) [1])
-- Reinhard Zumkeller, Apr 27 2012

readlib(issqr): f := i->if issqr(i) then 1 else 0; fi; [ seq(f(i),i=0..100) ];

lst = {}; Do[AppendTo[lst, 2*Sum[Floor[n/k] - Floor[(n - 1)/k], {k, Floor[Sqrt[n]]}] - DivisorSigma[0, n]], {n, 93}]; Prepend[lst, 1] (* Eric Desbiaux, Jan 29 2012 *)
Table[If[IntegerQ[Sqrt[n]],1,0],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
a[n_] := SeriesCoefficient[1/(1 - q)* QHypergeometricPFQ[{-q, -q}, {-(q^2)}, -q, -q], {q, 0, Abs@n}] (* Mats Granvik, Jan 01 2016 *)
Range[0, 120] /. {n_ /; IntegerQ@ Sqrt@ n -> 1, n_ /; n != 1 -> 0} (* Michael De Vlieger, Jan 02 2016 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[LiouvilleLambda[k]]*Sqrt[LiouvilleLambda[n/k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)

PARI
```
{a(n) = issquare(n)};
```

a(n)=if(n<1,1,sumdiv(n,d,(-1)^bigomega(d))) \\ Benoit Cloitre, Oct 25 2009

a(n) = if (n<1, 1, direuler( p=2, n, 1/ (1 - X^2 ))[n]); \\ Michel Marcus, Mar 08 2015

def A010052(n): return int(math.isqrt(n)**2==n) ##  appears to be faster than sympy.ntheory.primetest.is_square, up to 10^8 at least.
# M. F. Hasler, Mar 21 2022

(define (A010052 n) (if (zero? n) 1 (- (A000196 n) (A000196 (- n 1))))) ;; (For the definition of A000196, see under that entry). - Antti Karttunen, Nov 03 2017

a(n) = floor(sqrt(n)) - floor(sqrt(n-1)), for n > 0.
a(n) = A000005(n) mod 2, n > 0. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-w)^2 - (v-w)*(v+w-1) - Michael Somos, Jul 19 2004
Dirichlet g.f.: zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
G.f.: (theta_3(0,x) + 1)/2, where theta_3 is a Jacobi theta function. - Franklin T. Adams-Watters, Jun 19 2006 [See A000122 for theta_3.]
a(n) = f(n,0) with f(x,y) = f(x-2*y-1,y+1) if x > 0, otherwise 0^(-x). - Reinhard Zumkeller, Sep 26 2008
a(n) = Sum_{d|n} (-1)^bigomega(d), for n >= 1. - Benoit Cloitre, Oct 25 2009
a(n) <= A093709(n). - Reinhard Zumkeller, Nov 14 2009
a(A000290(n)) = 1; a(A000037(n)) = 0. - Reinhard Zumkeller, Jun 20 2011
a(n) = 0 ^ A053186(n). - Reinhard Zumkeller, Feb 12 2012
a(n) = A063524(A007913(n)), for n > 0. - Reinhard Zumkeller, Jul 09 2014
a(n) = -(-1)^n * A258998(n) unless n = 0. 2 * a(n) = A000122(n) unless n = 0. - Michael Somos, Jun 16 2015
a(n) = A037011(A156552(n)), provided that A037011(n) = A000035(A106737(n)). [See A037011.] - Antti Karttunen, Nov 03 2017
a(n*m) = a(n/gcd(n,m))*a(m/gcd(n,m)) for all n and m > 0 (conjectured). - Velin Yanev, Feb 13 2019 [Proof from Michael B. Porter, Feb 16 2019: If nm is a square, nm = product_i (p_i^2), where p_i are prime, not necessarily distinct. Each p_i either appears twice in n, twice in m, or one time in each and therefore in the gcd. So n/gcd(n,m) and m/gcd(n,m) are both squares. If nm is not a square, there is a q_j that appears in one of n or m but not in the gcd. So either n/gcd(n,m) or m/gcd(n,m) is not a square.]
a(n) = Sum_{d|n} A008836(d), n >= 1, a(0) = 1. - Jinyuan Wang, Apr 20 2019
G.f.: A(q) = Sum_{n >= 0}  q^(2*n)*Product_{k >= 2*n+1} 1 - (-q)^k. - Peter Bala, Feb 22 2021
Multiplicative with a(p^e) = 1 if e is even, and 0 otherwise. - Amiram Eldar, Dec 29 2022
a(n) = Sum_{d|n} mobius(core(n)), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

More terms from Franklin T. Adams-Watters, Jun 19 2006

A000161 Number of partitions of n into 2 squares.

Original entry on oeis.org

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

A025443 Number of partitions of n into 4 distinct nonzero squares.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Index entries for sequences related to sums of squares

Cf. A025428 (not necessarily distinct), A025376-A025394 (subsequences), A025417 (greedy inverse).

Column k=4 of A341040.

b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(t*i^2n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..150);  # Alois P. Heinz, Feb 07 2013

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[t*i^2n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 4]; Table[a[n], {n, 0, 150}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz*)
dnzs[n_]:=Length[Select[IntegerPartitions[n,{4}],Length[Union[#]]==4&&AllTrue[ Sqrt[ #], IntegerQ] && FreeQ[#,0]&]]; Array[dnzs,110,0] (* Harvey P. Dale, Jun 09 2024 *)

a(n) = [x^n y^4] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, May 30 2014

			T(20,5) = 2 = #{ (16,1,1,1,1), (4,4,4,4,4) } since 20 = 4^2 + 4 * 1^2 = 5 * 2^2.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  (...)

Alois P. Heinz, Rows n = 0..200, flattened

Columns k = 0..10 give: A000007, A010052 (for n>0), A025426, A025427, A025428, A025429, A025430, A025431, A025432, A025433, A025434.
Row sums give A001156.
T(2n,n) gives A111178.
T(n^2,n) gives A319435.
Cf. A000290, A276559, A281541, A292520, A341040.
T(n,k) = 1 for n in A025284, A025321, A025357, A294675, A295670, A295797 (for k = 2..7, respectively).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
T:= (n, k)-> b(n, isqrt(n), k):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(s-> `if`(s>n, 0, expand(x*b(n-s, i))))(i^2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, isqrt(n))):
seq(T(n), n=0..14);  # Alois P. Heinz, Oct 30 2021

b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i-1, k, t] + If[i^2 > n, 0, b[n-i^2, i, k, t-1]]]]; T[n_, k_] := b[n, Sqrt[n] // Floor, k, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 06 2014, after Alois P. Heinz *)
T[n_, k_] := Count[PowersRepresentations[n, k, 2], r_ /; FreeQ[r, 0]]; T[0, 0] = 1; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2016 *)

T(n,k,L=n)=if(n>k*L^2, 0, k>n-3, k==n, k<2, issquare(n,&n) && n<=L*k, k>n-6, n-k==3, L=min(L,sqrtint(n-k+1)); sum(r=0,min(n\L^2,k-1),T(n-r*L^2,k-r,L-1), n==k*L^2)) \\ M. F. Hasler, Aug 03 2020

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000290(j)).
Sum_{k=1..n} k * T(n,k) = A281541(n).
Sum_{k=1..n} n * T(n,k) = A276559(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A292520(n).

A025357 Numbers that are the sum of 4 nonzero squares in exactly 1 way.

Original entry on oeis.org

4, 7, 10, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 30, 33, 35, 38, 40, 44, 46, 48, 51, 53, 59, 62, 64, 65, 72, 80, 88, 89, 101, 104, 120, 152, 160, 176, 184, 192, 248, 256, 288, 320, 352, 416, 480, 608, 640, 704, 736, 768, 992, 1024, 1152, 1280, 1408, 1664

Offset: 1

David W. Wilson

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..138 (terms 1..95 from Robert Price)
Michael S. Branicky, Python program
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A025428, A243148.

selQ[n_] := Length[ Select[ PowersRepresentations[n, 4, 2], Times @@ # != 0 &]] == 1; Reap[Do[If[selQ[n], Print[n]; Sow[n]], {n, 1, 2000}]][[2, 1]] (* Jean-François Alcover, Oct 03 2013 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 4], {n, 0, 2000}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

# see link for faster version
limit = 1664
from functools import lru_cache
sq = [k*k for k in range(1, int(limit**.5)+2) if k*k + 3 <= limit]
sqs = set(sq)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n,)} if n in sqs else set()
  return set(tuple(sorted(t+(s, ))) for s in sq for t in findsums(n-s, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) == 1]) # Michael S. Branicky, Apr 07 2021

{n: A025428(n) = 1}. - R. J. Mathar, Jun 15 2018

A243148(a(n),4) = 1. - Alois P. Heinz, Feb 25 2019

A000534 Numbers that are not the sum of 4 nonzero squares.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 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

N. J. A. Sloane and J. H. Conway

For n > 15, a(n) = A006431(n-1). - Thomas Ordowski, Nov 18 2012

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 302.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, Theorem 3, pp. 74-75.

T. D. Noe, Table of n, a(n) for n = 1..100
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. See p. 2.
Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.
Index entries for linear recurrences with constant coefficients, signature (0,0,4).
Index entries for sequences related to sums of squares

Cf. A123069, A000414 (complement).

q=22;lst={};Do[Do[Do[Do[z=a^2+b^2+c^2+d^2;If[z<=q^2+3,AppendTo[lst,z]],{d,q}],{c,q}],{b,q}],{a,q}];lst1=Union@lst lst={};Do[AppendTo[lst,n],{n,q^2+3}];lst2=lst Complement[lst2,lst1] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Join[{0,1,2,3,5,6,8,9,11,14,17,24,29,32,41}, LinearRecurrence[{0, 0, 4}, {56, 96, 128}, 30]] (* Jean-François Alcover, Feb 09 2016 *)

for(n=1,224,if(sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,if(a^2+b^2+c^2+d^2-n,0,1)))))==0,print1(n,",")))

{a(n)=if( n<2, 0, n<16, [1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41][n-1], [4, 7, 12][n%3+1] * 2^(n\3*2-7))}; /* Michael Somos, Apr 23 2006 */

is(n)=my(k=if(n,n/4^valuation(n,4),2)); k==2 || k==6 || k==14 || setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014

from itertools import count, islice
def A000534_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2,6,14},count(max(startvalue,0)))
A000534_list = list(islice(A000534_gen(),30)) # Chai Wah Wu, Jul 09 2022

Consists of the numbers 0, 1, 3, 5, 9, 11, 17, 29, 41, 2*4^m, 6*4^m and 14*4^m (m >= 0). Compare A123069.
From 224 on, a(n) = 4*a(n-3).
Numbers n such that A025428(n) = 0.
G.f.: x^2*(36*x^16 + 32*x^15 + 60*x^14 + 55*x^13 + 36*x^12 + 27*x^11 + 20*x^10 + 19*x^9 + 18*x^8 + 13*x^7 + 11*x^6 + 4*x^5 + 2*x^4 - x^3 - 3*x^2 - 2*x - 1)/(4*x^3 - 1). - Chai Wah Wu, Jul 09 2022

A025367 Numbers that are the sum of 4 nonzero squares in 2 or more ways.

Original entry on oeis.org

28, 31, 34, 36, 37, 39, 42, 43, 45, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118

Offset: 1

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000414, A024796, A025358, A025368, A025406, A344795.

N:= 1000: # to get all terms <= N
V:= Vector(N):
for x from 1 while x^2 +3 <= N do
for y from 1 to x while x^2 + y^2 + 2 <= N do
  for z from 1 to y while x^2 + y^2 + z^2 + 1 <= N do
    for w from 1 to z while x^2 + y^2 + z^2 + w^2 <= N do
       t:= x^2 + y^2 + z^2 + w^2;
       V[t]:= V[t]+1;
od od od od:
select(t -> V[t] >= 2, [$1..N]); # Robert Israel, Jul 05 2017

Select[Range@ 200, 2 == Length@ Quiet@ IntegerPartitions[#, {4}, Range[Sqrt@ #]^2, 2] &] (* Giovanni Resta, Jul 05 2017 *)
M = 1000;
Clear[V]; V[_] = 0;
For[a = 1, a <= Floor[Sqrt[M/4]], a++,
  For[b = a, b <= Floor[Sqrt[(M - a^2)/3]], b++,
    For[c = b, c <= Floor[Sqrt[(M - a^2 - b^2)/2]], c++,
      For[d = c, d <= Floor[Sqrt[M - a^2 - b^2 - c^2]], d++,
        m = a^2 + b^2 + c^2 + d^2;
        V[m] = V[m] + 1;
]]]];
Select[Range[M], V[#] >= 2&] (* Jean-François Alcover, Mar 22 2019, after Robert Israel *)

{n: A025428(n) >= 2}. - R. J. Mathar, Jun 15 2018

A025371 Numbers that are the sum of 4 nonzero squares in 6 or more ways.

Original entry on oeis.org

90, 124, 130, 133, 135, 138, 147, 148, 150, 154, 156, 157, 159, 162, 163, 165, 166, 170, 171, 172, 174, 175, 177, 178, 180, 182, 183, 186, 187, 188, 189, 190, 193, 195, 196, 198, 199, 201, 202, 203, 205, 207, 210, 213, 214, 215, 217, 218, 219, 220, 222, 223, 225, 226

Offset: 1

David W. Wilson

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A025334, A025362, A025370, A025372, A344799, A345148.

limit = 226
from functools import lru_cache
sq = [k**2 for k in range(1, int(limit**.5)+2) if k**2 + 3 <= limit]
sqs = set(sq)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n, )} if n in sqs else set()
  return set(tuple(sorted(t+(s,))) for s in sqs for t in findsums(n-s, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 6]) # Michael S. Branicky, Apr 20 2021

{n: A025428(n) >= 6}. Union of A025372 and A025362. - R. J. Mathar, Jun 15 2018

A025416 Least sum of 4 nonzero squares in exactly n ways.

Original entry on oeis.org

0, 4, 31, 28, 52, 82, 90, 135, 130, 162, 198, 202, 252, 234, 210, 346, 306, 322, 423, 370, 330, 418, 390, 462, 378, 490, 598, 450, 546, 618, 522, 594, 642, 682, 570, 770, 714, 690, 762, 906, 738, 630, 1030, 850, 1035, 978, 858, 954, 810, 1197, 1146, 882, 1090, 1206

Offset: 0

David W. Wilson

nonn

Conjecture: The sequence never becomes monotonic increasing. - Jon Perry, Nov 03 2012

			a(2) = 31 because 31 = 1 + 1 + 4 + 25 = 4 + 9 + 9 + 9 and no others.
a(3) = 28 because 28 = 1 + 1 + 1 + 25 = 1 + 9 + 9 + 9 = 4 + 4 + 4 + 16 and no others.
a(4) = 52 because 52 = 1 + 1 + 1 + 49 = 1 + 1 + 25 + 25 = 4 + 16 + 16 + 16 = 9 + 9 + 9 + 25 and no others.

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 501 terms from T. D. Noe)
Index entries for sequences related to sums of squares

nn = 40; t = Select[Flatten[Table[a^2 + b^2 + c^2 + d^2, {a, nn}, {b, a}, {c, b}, {d, c}]], # <= nn^2 + 3 &]; {t1, t2} = Transpose[Sort[Tally[t]]]; u = Union[t2]; c = Complement[Range[u[[-1]]], u]; If[c == {}, last = u[[-1]], last = c[[1]] - 1]; Join[{0}, Table[t1[[Position[t2, n, 1, 1][[1, 1]]]], {n, last}]] (* T. D. Noe, Nov 02 2012 *)

{min k: A025428(k) = n}. - R. J. Mathar, Jun 15 2018

0th term added by Jon Perry, Nov 02 2012

cp's OEIS Frontend

A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

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A000161 Number of partitions of n into 2 squares.

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A025443 Number of partitions of n into 4 distinct nonzero squares.

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A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.

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A025357 Numbers that are the sum of 4 nonzero squares in exactly 1 way.

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A000534 Numbers that are not the sum of 4 nonzero squares.

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