A003995 -id:A003995 - cp's OEIS frontend

A033461 Number of partitions of n into distinct squares.

1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 2, 3, 1, 1, 4, 3, 0, 1, 2, 2, 1, 0, 1, 4, 3, 0, 2, 4, 2, 1, 3, 2, 1, 2, 3, 3, 2, 1, 3, 6, 3, 0, 2, 5, 3, 0, 1, 3, 3, 3, 4

Offset: 0

N. J. A. Sloane

"WEIGH" transform of squares A000290.
a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}. - Alois P. Heinz, May 14 2014
Number of partitions of n in which each part i has multiplicity i. Example: a(50)=3 because we have [1,2,2,3,3,3,6,6,6,6,6,6], [1,7,7,7,7,7,7,7], and [3,3,3,4,4,4,4,5,5,5,5,5]. - Emeric Deutsch, Jan 26 2016
The Heinz numbers of integer partitions into distinct pairs are given by A324587. - Gus Wiseman, Mar 09 2019
From Gus Wiseman, Mar 09 2019: (Start)
Equivalent to Emeric Deutsch's comment, a(n) is the number of integer partitions of n where the multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in increasing order. The Heinz numbers of these partitions are given by A109298. For example, the first 30 terms count the following integer partitions:
(1)
(22)
(221)
(333)
(3331)
(33322)
(333221)
(4444)
(44441)
(444422)
(4444221)
(55555)
(4444333)
(555551)
(44443331)
(5555522)
(444433322)
(55555221)
(4444333221)
The case where the distinct parts are taken in decreasing order is A324572, with Heinz numbers given by A324571.
(End)

			a(50)=3 because we have [1,4,9,36], [1,49], and [9,16,25]. - _Emeric Deutsch_, Jan 26 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
The first 30 terms count the following integer partitions:
   1: (1)
   4: (4)
   5: (4,1)
   9: (9)
  10: (9,1)
  13: (9,4)
  14: (9,4,1)
  16: (16)
  17: (16,1)
  20: (16,4)
  21: (16,4,1)
  25: (25)
  25: (16,9)
  26: (25,1)
  26: (16,9,1)
  29: (25,4)
  29: (16,9,4)
  30: (25,4,1)
  30: (16,9,4,1)
(End)

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 288-289.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Brack, M. V. N. Murthy, and J. Bartel, Application of semiclassical methods to number theory, University of Regensburg (Germany, 2020).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, Graph - The asymptotic ratio.
M. V. N. Murthy, Matthias Brack, Rajat K. Bhaduri, and Johann Bartel, Semi-classical analysis of distinct square partitions, arXiv:1808.05146 [cond-mat.stat-mech], 2018.

Cf. A001422, A003995, A078434, A242434 (the same for compositions), A279329.
Cf. A001156 (non-strict case), A001462, A005117, A052335, A078135, A109298, A114638, A117144, A324571, A324572, A324587, A324588.
Row sums of A341040.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i-1))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 14 2014

nn=10; CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x] (* T. D. Noe, Jul 24 2006 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n - i^2, i-1]]]]; a[n_] := b[n, Floor[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, Dec 09 2016 *)
Table[Length[Select[IntegerPartitions[n],Reverse[Union[#]]==Length/@Split[#]&]],{n,30}] (* Gus Wiseman, Mar 09 2019 *)

a(n)=polcoeff(prod(k=1,sqrt(n),1+x^k^2), n)

first(n)=Vec(prod(k=1,sqrtint(n),1+'x^k^2,O('x^(n+1))+1)) \\ Charles R Greathouse IV, Sep 03 2015

from functools import cache
from sympy.core.power import isqrt
@cache
def b(n,i):
  # Code after Alois P. Heinz
  if n == 0: return 1
  if i == 0: return 0
  i2 = i*i
  return b(n, i-1) + (0 if i2 > n else b(n - i2, i-1))
a = lambda n: b(n, isqrt(n))
print([a(n) for n in range(1, 101)]) # Darío Clavijo, Nov 30 2023

G.f.: Product_{n>=1} ( 1+x^(n^2) ).
a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * ((sqrt(2)-1)*zeta(3/2))^(2/3) * n^(1/3)) * ((sqrt(2)-1)*zeta(3/2))^(1/3) / (2^(4/3) * sqrt(3) * Pi^(1/3) * n^(5/6)), where zeta(3/2) = A078434. - Vaclav Kotesovec, Dec 09 2016
See Murthy, Brack, Bhaduri, Bartel (2018) for a more complete asymptotic expansion. - N. J. A. Sloane, Aug 17 2018

More terms from Michael Somos

A001422 Numbers which are not the sum of distinct squares.

Original entry on oeis.org

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128

Offset: 1

N. J. A. Sloane, Jeff Adams (jeff.adams(AT)byu.net)

This is the complete list (Sprague).

S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 222.

R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
T. Sillke, Not the sum of distinct squares
R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z. 51, (1948), 289-290.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Complement of A003995. Subsequence of A004441.
Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A121405 (corresponding sequences for triangular and pentagonal numbers)
Cf. A033461, A276517.
Cf. A001476, A046039, A194768, A194769 for 3rd, 4th, 5th, 6th powers.
Cf. A001661.

nn=50; t=Rest[CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

select( is_A001422(n,m=n)={m^2>n&& m=sqrtint(n); n!=m^2&&!while(m>1,isSumOfSquares(n-m^2,m--)&&return)}, [1..128]) \\ M. F. Hasler, Apr 21 2020

A004432 Numbers that are the sum of 3 distinct nonzero squares.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 56, 59, 61, 62, 65, 66, 69, 70, 74, 75, 77, 78, 81, 83, 84, 86, 89, 90, 91, 93, 94, 98, 101, 104, 105, 106, 107, 109, 110, 113, 114, 115, 116, 117, 118, 120, 121, 122, 125, 126, 129, 131, 133

Offset: 1

N. J. A. Sloane

Numbers that can be written as a(n) = x^2 + y^2 + z^2 with 0 < x < y < z.
This is a subsequence (equal to the range) of A024803. As a set, it is the union of A025339 and A024804, subsequences of numbers having exactly one, resp. more than one, such representations. - M. F. Hasler, Jan 25 2013
Conjecture: a number n is a sum of 3 squares, but not a sum of 3 distinct nonzero squares (i.e., is in A004432 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 22, 25, 27, 33, 34, 37, 43, 51, 57, 58, 67, 73, 82, 85, 97, 99, 102, 123, 130, 163, 177, 187, 193, 267, 627, 697}. - Jeffrey Shallit, Jan 15 2017
4*a(n) gives the sums of 3 distinct nonzero even squares. - Wesley Ivan Hurt, Apr 05 2021

			14 = 1^2 + 2^2 + 3^2;
62 = 1^2 + 5^2 + 6^2.

Cf. A001974, A024803, A025339, A025442.

Cf. A000290, A003995, A004431, A004433, A004434, A224981, A224982, A224983.

a004432 n = a004432_list !! (n-1)
a004432_list = filter (p 3 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

f[upto_]:=Module[{max=Floor[Sqrt[upto]]},Select[Union[Total/@ (Subsets[ Range[ max],{3}]^2)],#<=upto&]]; f[150]  (* Harvey P. Dale, Mar 24 2011 *)

is_A004432(n)=for(x=1,sqrtint(n\3),for(y=x+1,sqrtint((n-1-x^2)\2),issquare(n-x^2-y^2)&return(1)))  \\ M. F. Hasler, Feb 02 2013

A004432 = { x^2 + y^2 + z^2; 0 < x < y < z }.

n is in A004432 <=> A025442(n) > 0. - M. F. Hasler, Feb 03 2013

A004433 Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0

Original entry on oeis.org

30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137

Offset: 1

N. J. A. Sloane

			30 = 1^2+2^2+3^2+4^2.

Cf. A001944, A001995, A003995, A004431, A004432, A004434, A224981, A224982, A224983, A000290.

a004433 n = a004433_list !! (n-1)
a004433_list = filter (p 4 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

data = Flatten[ DeleteCases[ FindInstance[ w^2 + x^2 + y^2 + z^2 == # && 0 < w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[137], {}], 1]; w^2 + x^2 + y^2 + z^2 /. data (* Ant King, Oct 17 2010 *)
Select[Union[Total[#^2]&/@Subsets[Range[10],{4}]],#<=137&] (* Harvey P. Dale, Jul 03 2011 *)

list(lim)=my(v=List([30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 149, 150, 151, 153, 154, 155, 156]), u=[160, 168, 172, 176, 188, 192, 208, 220, 224, 232, 240, 256, 268, 272, 288, 292, 304, 320, 328, 352, 368, 384, 388, 400, 412, 416, 432, 448, 496, 512, 528, 544, 576, 592, 608], t=1); if(lim<156, return(select(k->k<=lim, Vec(v)))); for(n=158,lim\1, if(n#u, t=1)); Vec(v) \\ Charles R Greathouse IV, Jan 08 2025

{n: A025443(n) >=1}. Union of A025386 and A025376. - R. J. Mathar, Jun 15 2018

a(n) = n + O(log n). - Charles R Greathouse IV, Jan 08 2025

A097563 Least integer that can be written as a sum of zero or more distinct squares in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

2, 0, 25, 50, 65, 94, 90, 110, 155, 126, 191, 170, 186, 174, 190, 211, 195, 226, 210, 231, 234, 235, 332, 255, 283, 259, 274, 275, 270, 323, 310, 286, 306, 299, 330, 381, 295, 347, 334, 319, 315, 331, 405, 339, 335, 364, 359, 351, 367, 387, 371, 370, 404, 438

Offset: 0

Isabel C. Lugo (izzycat(AT)gmail.com), Aug 27 2004

a(n) = -1 for almost all n. Conjecture: for n > 34189857569982621, this sequence is the integers > 37163, in order, interspersed with -1s. - Charles R Greathouse IV, Sep 04 2015

			a(4) = 65 because we can write 65 as a sum of distinct squares in four ways: 65 = 8^2 + 1^2 = 7^2 + 4^2 = 6^2 + 5^2 + 2^2 = 6^2 + 4^2 + 3^2 + 2^2 and we cannot do this with any smaller integer.
a(0) = 2 because we cannot write 2 as a sum of distinct squares and it is the smallest number with this property.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

First occurrence of n in A033461; see also A001422 (0 ways) and A003995 (1 or more ways).

gf := product(1+x^F(k), k=1..31); ser := series(gf, x=0, 1001); S := [seq(coeff(ser,x^(1*i)),i=1..1000)]; A := proc(i); x := 0; for j from 1 to nops(a) while x = 0 do > if a[j] = i then x := 1; fi; od; j-1; end; seq(A(n), n=1..67);

Edited by Ray Chandler, Sep 01 2004

A003997 Sums of distinct positive cubes.

Original entry on oeis.org

1, 8, 9, 27, 28, 35, 36, 64, 65, 72, 73, 91, 92, 99, 100, 125, 126, 133, 134, 152, 153, 160, 161, 189, 190, 197, 198, 216, 217, 224, 225, 243, 244, 251, 252, 280, 281, 288, 289, 307, 308, 315, 316, 341, 342, 343, 344, 349, 350, 351, 352, 368, 369, 370, 371

Offset: 1

N. J. A. Sloane

12758 is the largest of 2788 positive integers not in this sequence. - Jud McCranie, Dec 11 1999

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 12758.

T. D. Noe, Table of n, a(n) for n = 1..10000
R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51, (1948), 466-468.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Complement of A001476. Cf. A003995.

GF := series( (1+x)*(1+x^8)*(1+x^27)*(1+x^64)*(1+x^125)*(1+x^216)*(1+x^343)*(1+x^512)*(1+x^729)*(1+x^1000), x, 11^3); # Edited by M. F. Hasler, May 01 2020
A003997_upto := n -> map(degree,{op(convert(series(product(1 + x^(k^3), k = 1 .. floor(root(n,3)))-1, x, n+1),`+`))});  # M. F. Hasler, May 01 2020;

lim = 8; s = {0}; Do[s = Union[s, s + n^3], {n, lim}]; Select[s, 0 < # <= lim^3 &] (* T. D. Noe, Jul 10 2012 *)

list(lim)={
    lim\=1;
    my(lm=min(lim+1,12758), v=List(), P);
    P=prod(n=1,lm^(1/3),1+x^(n^3),1+O(x^lm));
    for(n=1,lm-1,if(polcoeff(P,n),listput(v,n)));
    if(lim>12758,concat(Vec(v),vector(lim-12758,i,i+12758)),Vec(v))
}; \\ Charles R Greathouse IV, Sep 02 2011

select( is_A003997(n,m=n)={m^3>n&&m=sqrtnint(n,3);n==m^3||while(m>1,is_A003997(n-m^3,m--)&&return(1))}, [1..400]) \\ M. F. Hasler, Apr 21 2020

For n > 9970, a(n) = n + 2788. - Charles R Greathouse IV, Sep 02 2011

Definition clarified by Jeppe Stig Nielsen, Jan 27 2015

A004434 Numbers that are the sum of 5 distinct nonzero squares.

Original entry on oeis.org

55, 66, 75, 79, 82, 87, 88, 90, 94, 95, 99, 100, 103, 106, 110, 111, 114, 115, 118, 120, 121, 123, 126, 127, 129, 130, 131, 132, 134, 135, 138, 139, 142, 143, 144, 145, 146, 147, 148, 150, 151, 152, 154, 155, 156, 157, 158, 159, 160, 162, 163

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003995, A004431, A004432, A004433, A224981, A224982, A224983, A000290.

a004434 n = a004434_list !! (n-1)
a004434_list = filter (p 5 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

upto(lim)=my(v=List(), tb, tc, td, te); for(a=5, sqrt(lim), for(b=4, min(a-1, sqrt(lim-a^2)), tb=a^2+b^2; for(c=3, min(b-1, sqrt(lim-tb)), tc=tb+c^2; for(d=2, min(c-1, sqrt(lim-tc)), td=tc+d^2; for(e=1, d-1, te=td+e^2; if(te>lim, break,listput(v, te))))))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jul 17 2011

a(n) = n + 124 for n > 121. - Charles R Greathouse IV, Jul 17 2011

A224981 Numbers that are the sum of exactly 6 distinct nonzero squares.

Original entry on oeis.org

91, 104, 115, 119, 124, 130, 131, 136, 139, 143, 146, 147, 151, 152, 154, 155, 156, 159, 160, 163, 164, 166, 167, 168, 169, 170, 171, 175, 176, 178, 179, 180, 181, 182, 184, 187, 188, 190, 191, 192, 194, 195, 196, 199, 200, 201, 202, 203, 204, 206, 207, 208

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 = 91 = A000330(6);
a(2) = 1 + 4 + 9 + 16 + 25 + 49 = 104;
a(3) = 1 + 4 + 9 + 16 + 36 + 49 = 115;
a(4) = 1 + 4 + 9 + 16 + 25 + 64 = 119;
a(5) = 1 + 4 + 9 + 25 + 36 + 49 = 124.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Index entries for sequences related to sums of squares

Cf. A003995, A004431, A004432, A004433, A004434, A224982, A224983, A000290.

a224981 n = a224981_list !! (n-1)
a224981_list = filter (p 6 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

nmax = 1000;
S[n_] := S[n] = Union[Total /@ Subsets[
     Range[Floor[Sqrt[n]]]^2, {6}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

A224982 Numbers that are the sum of exactly 7 distinct nonzero squares.

Original entry on oeis.org

140, 155, 168, 172, 179, 185, 188, 191, 195, 196, 200, 203, 204, 205, 211, 212, 215, 217, 219, 220, 224, 225, 227, 230, 231, 232, 233, 235, 236, 239, 240, 243, 244, 245, 246, 247, 248, 251, 252, 254, 256, 257, 259, 260, 263, 264, 265, 267, 268, 269, 270, 271

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140 = A000330(7);
a(2) = 1 + 4 + 9 + 16 + 25 + 36 + 64 = 155;
a(3) = 1 + 4 + 9 + 16 + 25 + 49 + 64 = 168;
a(4) = 1 + 4 + 9 + 16 + 25 + 36 + 81 = 172;
a(5) = 1 + 4 + 9 + 16 + 36 + 49 + 64 = 179.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Index entries for sequences related to sums of squares

Cf. A003995, A004431, A004432, A004433, A004434, A224981, A224983, A000290.

a224982 n = a224982_list !! (n-1)
a224982_list = filter (p 7 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

nmax = 1000;
S[n_] := S[n] = Union[Total /@ Subsets[
     Range[Floor[Sqrt[n]]]^2, {7}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

cp's OEIS Frontend

A178132 Partial sums of A003995.

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A033461 Number of partitions of n into distinct squares.

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A001422 Numbers which are not the sum of distinct squares.

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A004432 Numbers that are the sum of 3 distinct nonzero squares.

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A004433 Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0

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A097563 Least integer that can be written as a sum of zero or more distinct squares in exactly n ways, or -1 if no such number exists.

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A003997 Sums of distinct positive cubes.

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