A004433 -id:A004433 - cp's OEIS frontend

A316833 Sums of four distinct odd squares.

84, 116, 140, 156, 164, 180, 196, 204, 212, 228, 236, 244, 252, 260, 276, 284, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 396, 404, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548, 556, 564, 572, 580, 588, 596, 604, 612, 620, 628, 636, 644, 652, 660

Offset: 1

N. J. A. Sloane, Jul 19 2018

nonn

Theorem (Conjectured by R. William Gosper, proved by M. D. Hirschhorn): Any sum of four distinct odd squares is the sum of four distinct even squares.
The proof uses the following identity:
(4a+1)^2+(4b+1)^2+(4c+1)^2+(4d+1)^2 = 4[ (a+b+c+d+1)^2 + (a-b-c+d)^2 + (a-b+c-d)^2 + (a+b-c-d)^2 ].
All terms == 4 (mod 8).  Are all numbers == 4 (mod 8) and > 412 members of the sequence? - Robert Israel, Jul 20 2018

R. William Gosper and Stephen K. Lucas, Postings to Math Fun Mailing List, July 19 2018
Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.

Robert Israel, Table of n, a(n) for n = 1..10000

A316834 lists the subsequence for which the representation is unique.

Cf. A004433, A316835.

N:= 1000: # to get all terms <= N
V:= Vector(N):
for a from 1 to floor(sqrt(N/4)) by 2 do
  for b from a+2 to floor(sqrt((N-a^2)/3)) by 2 do
    for c from b+2 to floor(sqrt((N-a^2-b^2)/2)) by 2 do
      for d from c + 2  by 2 do
        r:= a^2+b^2+c^2+d^2;
        if r > N then break fi;
        V[r]:= V[r]+1
od od od od:
select(t -> V[t]>=1, [$1..N]); # Robert Israel, Jul 20 2018

A175958 Number of partitions of n^2 into 4 distinct nonzero squares.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 5, 0, 4, 4, 5, 0, 10, 4, 7, 0, 11, 7, 17, 1, 13, 17, 15, 0, 29, 13, 27, 4, 23, 17, 41, 0, 29, 35, 32, 4, 66, 24, 38, 0, 47, 35, 73, 7, 50, 56, 73, 1, 91, 42, 63, 17, 68, 49, 125, 0, 103, 93, 83, 13, 133, 86, 93, 4

Offset: 0

R. J. Mathar, Oct 30 2010

			a(9)=1 refers to the partition 9^2 = 2^2+4^2+5^2+6^2. a(11)=1 refers to 11^2 = 1^2+2^2+4^2+10^2. a(13)=2 refers to 13^2 = 1^2+2^2+8^2+10^2 = 2^2+4^2+7^2+10^2.

Alois P. Heinz and Donovan Johnson, Table of n, a(n) for n = 0..1000 (terms up to a(300) from Alois P. Heinz)

Cf. A004195, A004441, A004433, A181524.

A025443 := proc(n) local res,a,b,c,d ; res := 0 ; for a from 1 do if 4*a^2 > n then break; fi; for b from a+1 do if a^2+3*b^2 > n then break; fi; for c from b+1 do if a^2+b^2+2*c^2 > n then break; fi; for d from c+1 do if a^2+b^2+c^2+d^2 > n then break; elif a^2+b^2+c^2+d^2 = n then res := res+1 ; fi ; end do; end do; end do: end do: res ; end proc:
A := proc(n) A025443(n^2) ; end proc: seq(A(n),n=0..60) ;
# second Maple program:
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^2, n, 4):
seq(a(n), n=0..80);  # 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^2 < n, 0, If[i == 1, 0, b[n, i-1, t]] + If[i^2 > n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n^2, n, 4]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)

a(n) = A025443(n^2).

More terms from Alois P. Heinz, Feb 07 2013

A223727 Numbers which are a sum of four distinct nonzero squares where the summands have no common factor > 1.

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, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140

Offset: 1

Wolfdieter Lang, Mar 27 2013

nonn

A primitive representation of a number m as a sum of four distinct nonzero squares is determined from a quadruple [s(1), s(2), s(3), s(4)] of integers with 0 < s(1) < s(2) < s(3) < s(4) with gcd(s(1),s(2),s(3),s(4)) = 1, and m = sum(s(j)^2, j=1..4). If m has such a primitive representation then k^2*m, with integer k > 0, has trivially a non-primitive representation. Therefore primitive representations are of interest.
For the multiplicities see A223728.
This sequence is a proper subset of A004433. The first entry of A004433 missing here is 120 = A004433(43). The first common entry with different multiplicity is A004433(72) = 156 = a(71) with two primitive representations with quadruples
[1, 3, 5, 11] and [1, 5, 7, 9]. [2, 4, 6, 10] = 2*[1, 2, 3, 5]is a non-primitive representation due to 156 = 4*39.

			a(1) = 30 because the numbers 0,...,29 have no representation as a sum of four distinct nonzero squares, and 30 has one representation given by the quadruple [1,2,3,4] which is primitive.
a(16) = 78 has three such representations given by the quadruples  [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7] which are all primitive. Hence A223728(16) = 3. This is the first entry with more than one (primitive) representation.
a(23) = 90 has multiplicity 2 = A223728 because there are two primitive quadruples [1, 2, 6, 7] and [1, 3, 4, 8].
a(71) = 156 has multiplicity A223728(71) = 2 (see a comment above).

Cf. A222949, A097203, A223728, A259058 (multiplicity >= 2 instances).

This sequence are the increasingly ordered members of the set {m an integer | m = sum(s(j)^2, j=1..4), with 0 < s(1) < s(2) < s(3) < s(4) and gcd(s(1),s(2),s(3),s(4)) = 1}.

A316835 Sums of four distinct positive even squares.

Original entry on oeis.org

120, 156, 184, 200, 204, 216, 228, 248, 252, 260, 264, 280, 284, 296, 300, 312, 316, 324, 336, 340, 344, 348, 360, 364, 372, 376, 380, 392, 396, 408, 420, 424, 428, 436, 440, 444, 452, 456, 464, 468, 472, 476, 480, 484, 488, 492, 500, 504, 508, 516, 520, 524, 532, 536, 540, 548, 552

Offset: 1

N. J. A. Sloane, Jul 19 2018

nonn

Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.

Equals 4*A004433. Cf. A316833, A316834.

Total/@Subsets[(2*Range[10])^2,{4}]//Union (* Harvey P. Dale, May 21 2019 *)

A178096 Cube of n is equal to sum of four positive distinct squares; n^3=a^2+b^2+c^2+d^2; a>b>c>d>0.

Original entry on oeis.org

5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 19 2010

nonn

5^3=8^2+6^2+4^2+3^2, 6^3=10^2+8^2+6^2+4^2, ...

Cf. A003294, A005767, A023809, A046100.

z=100;lst={};Do[a2=a^2;Do[b2=b^2;Do[c2=c^2;Do[d2=d^2;e2=a2+b2+c2+d2;e=e2^(1/3);If[IntegerQ[e],AppendTo[lst,e]],{d,c-1,1,-1}],{c,b-1,1,-1}],{b,a-1,1,-1}],{a,1,z}];Union@lst

{n: n^3 in A004433}. - R. J. Mathar, Jun 15 2018

Terms > 33 from R. J. Mathar, Jun 15 2018

A375744 Numbers that are the sum of 4 but no fewer nonzero squares and admit a representation with 4 distinct squares.

Original entry on oeis.org

39, 63, 71, 79, 87, 95, 111, 119, 127, 135, 143, 151, 156, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 252, 255, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343, 348, 351, 359, 367, 375, 380, 383, 391, 399, 407, 415, 423, 431

Offset: 1

Gonzalo Martínez, Aug 26 2024

nonn

Intersection of A004215 and A004433.

			39 is a term, since it requires 4 squares to be represented and admits the representation 39 = 5^2 + 3^2 + 2^2 + 1^2.
30 is not a term, although it can be represented as a sum of 4 different squares 30 = 4^2+ 3^2 + 2^2 + 1^2 also admits a representation as a sum of 3 squares: 30 = 5^2 + 2^2 + 1^2.
7 is not a term, since although it requires 4 squares to be represented as follows 7 = 2^2 + 1^2 + 1^2 + 1^2, it is noted that 1 is used on more than one occasion.

Cf. A004215, A004433.

cp's OEIS Frontend

A316833 Sums of four distinct odd squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

A175958 Number of partitions of n^2 into 4 distinct nonzero squares.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A223727 Numbers which are a sum of four distinct nonzero squares where the summands have no common factor > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A316835 Sums of four distinct positive even squares.

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

A178096 Cube of n is equal to sum of four positive distinct squares; n^3=a^2+b^2+c^2+d^2; a>b>c>d>0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A375744 Numbers that are the sum of 4 but no fewer nonzero squares and admit a representation with 4 distinct squares.

Views

Author

Keywords

Comments

Examples

Crossrefs