A004433 -id:A004433 - cp's OEIS frontend

A223726 Multiplicities for A004433: sum of four distinct nonzero squares.

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

Offset: 1

Wolfdieter Lang, Mar 26 2013

nonn

The number A004433(n) can be partitioned into four distinct parts, each of which is a nonzero square, and a(n) gives the multiplicity which is the number of different partitions of this type.

			a(1) = 1 because  A004433(1) = 30 has only one representation as sum of four distinct nonzero squares, given by the quadruple [1,2,3,4]: 1^2 + 2^2 + 3^2 + 4^2 = 30.
a(16) = 3 because for A004433(3) = 78 the three different quadruples are [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7].
a(48) = 5 because A004433(48) = 126 has five different  representations given by the five quadruples [1, 3, 4, 10], [1, 5, 6, 8], [2, 3, 7, 8], [2, 4, 5, 9], [4, 5, 6, 7].

Cf. A004433, A025428, A000414, A097203, A222949.

a(n) = k if there are k different quadruples [s(1),s(2),2(3),s(4)] with increasing positive entries with sum(s(j)^2,j=1..4) = A004433(n), n >= 1.

A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

Original entry on oeis.org

0, 1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 25, 26, 29, 30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001983, A033461, A008935. Complement of A001422.

Cf. A000290; subsequences: A004431, A004432, A004433, A004434, A224981, A224982, A224983.

a003995 n = a003995_list !! (n-1)
a003995_list = filter (p a000290_list) [0..]
   where p (q:qs) m = m == 0 || q <= m && (p qs (m - q) || p qs m)
-- Reinhard Zumkeller, Apr 22 2013

lim = 10; s = {0}; Do[s = Union[s, s + n^2], {n, lim}]; Select[s, 0 <= # <= lim^2 &] (* T. D. Noe, Jul 10 2012 *)

a(n)=if(n<1,0,n=a(n-1); until(polcoeff(prod(k=1,sqrt(n),1+x^k^2), n), n++); n)

Exponents in expansion of (1+x)*(1+x^4)*(1+x^9)*(1+x^16)*(1+x^25)*(1+x^36)*(1+x^49)*(1+x^64)*(1+x^81)*(1+x^100)*(1+x^121)*(1+x^144)*...

For n > 98, a(n) = n + 30. - Charles R Greathouse IV, Sep 02 2011 (This implies a(n+2) = 2*a(n+1)-a(n) for n > 98.)

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

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 *)

A224983 Numbers that are the sum of exactly 8 distinct nonzero squares.

Original entry on oeis.org

204, 221, 236, 240, 249, 255, 260, 261, 268, 269, 272, 276, 279, 281, 284, 285, 288, 289, 293, 295, 296, 299, 300, 303, 305, 306, 309, 311, 312, 316, 317, 320, 321, 323, 324, 325, 326, 327, 329, 332, 333, 335, 336, 337, 338, 339, 340, 341, 344, 345, 347, 348

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204 = A000330(8);
a(2) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 81 = 221;
a(3) = 1 + 4 + 9 + 16 + 25 + 36 + 64 + 81 = 236;
a(4) = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 100 = 240;
a(5) = 1 + 4 + 9 + 16 + 25 + 49 + 64 + 81 = 249.

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, A224982, A000290.

a224983 n = a224983_list !! (n-1)
a224983_list = filter (p 8 $ 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, {8}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

A004441 Numbers that are not the sum of 4 distinct nonzero squares.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 52, 53, 55, 56, 58, 59, 60, 61, 64, 67, 68

Offset: 1

N. J. A. Sloane

It has been shown that 157 is the last odd number in this sequence. Beyond 157, the terms grow exponentially. - T. D. Noe, Apr 07 2007

Taking a(86) to a(120) as initial terms, A004441(n) satisfies the 35th-order recurrence relation u(n) = 4*u(n-35). - Ant King, Aug 13 2010

T. D. Noe, Table of n, a(n) for n = 1..1000 (using A004195 and A004196)
Paul T. Bateman, Adolf J. Hildebrand and George B. Purdy, Sums of distinct squares, Acta Arith. 67 (1994), 349-380.
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. [Mentions this sequence.]
Index entries for linear recurrences with constant coefficients, signature (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, 0, 0, 0, 0, 4).
Index entries for sequences related to sums of squares

Cf. A004195, A004196, A004433 (complement).

data1=Reduce[w^2+x^2+y^2+z^2==# && 00,0,k],{k,1,Length[data2]}],0] (* Ant King, Aug 13 2010 *)

A033985 Number of partitions of n into two or more distinct nonzero squares.

Original entry on oeis.org

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

Offset: 1

Jeff Burch

nonn

Index entries for sequences related to sums of squares

Cf. A033461, A004431, A004432, A004433, A004434.

G.f.: Product_{k>=1} (1+x^(k^2)) - Sum_{k>=1} x^(k^2). - Sean A. Irvine, Jul 26 2020

Title improved by Sean A. Irvine, Jul 26 2020

A259058 Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.

Original entry on oeis.org

454, 530, 614, 706, 806, 914, 1030, 1154, 1286, 1426, 1574, 1730, 1894, 2066, 2246, 2434, 2630, 2834, 3046, 3266, 3494, 3730, 3974, 4226, 4486, 4754, 5030, 5314, 5606, 5906, 6214, 6530, 6854, 7186, 7526, 7874, 8230, 8594, 8966, 9346, 9734

Offset: 0

Wolfdieter Lang, Aug 12 2015

This is part one of Exercise 229 in Sierpiński's problem book. See p. 20 and p. 110 for the solution. He uses the identity (n-8)^2 + (n-1)^2 + (n+1)^2 + (n+8)^2 = 4*n^2 + 130 = (n-7)^2 + (n-4)^2 + (n+4)^2 + (n+7)^2, for n >= 9.
  Here n was replaced by n + 9: (n+1)^2 + (n+8)^2 +(n+10)^2 + (n+17)^2 = 4*n^2 + 72*n + 454 = (n+2)^2 + (n+5)^2 + (n+13)^2 + (n+16)^2, for n >= 0.
There may be other numbers having this property.
Because the summands have no common factor > 1 each of these two representations is called primitive. Therefore, this is a proper subsequence of A223727, hence of A004433. - Wolfdieter Lang, Aug 20 2015

			n=0: 454 = 1^2 + 8^2 + 10^2 + 17^2 = 2^2 + 5^2 + 13^2 + 16^2.
n=2: 614 = 3^2 + 10^2 + 12^2 + 19^2 = 4^2 + 7^2 + 15^2 + 18^2.

W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A259059, A223727, A004433, A259060 (four cubes).

[4*n^2 + 72*n + 454: n in [0..50]]; // Vincenzo Librandi, Aug 13 2015

I:=[454, 530, 614]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 13 2015

CoefficientList[Series[2 (227 - 416 x + 193 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 13 2015 *)

a(n)=4*n^2+72*n+454 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*n^2 + 72*n + 454 = 2*A259059(n). See the comment for the sum of four squares in two ways.
O.g.f.: 2*(227 - 416*x + 193*x^2)/(1-x)^3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Vincenzo Librandi, Aug 13 2015

cp's OEIS Frontend

A223726 Multiplicities for A004433: sum of four distinct nonzero squares.

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A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

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

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

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

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

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A224983 Numbers that are the sum of exactly 8 distinct nonzero squares.

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A004441 Numbers that are not the sum of 4 distinct nonzero squares.

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