A122922 -id:A122922 - cp's OEIS frontend

A122921 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, minimizing the value of x. a(n) is that x.

0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 4, 4, 5, 4, 4, 4, 4, 5, 4, 5, 5, 4, 4, 4, 5, 4, 6, 5, 5, 6, 4, 5, 5, 5, 5, 6, 5, 4, 6, 5, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 6, 6, 5, 6, 6, 5, 7, 5, 6, 6, 6, 6, 6, 6, 5, 6, 6, 7, 6, 8, 6, 6, 7, 5, 6, 6, 7, 6

Offset: 0

Franklin T. Adams-Watters, Sep 19 2006

nonn

			10 = 2^2 + 2^2 + 1^2 + 1^2, so a(10) = 2. The only representation for 11 is 3^2 + 1^2 + 1^2 + 0^2, so a(11) = 3.

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python program

Cf. A122922, A122923, A122924, A122925, A122926, A122927, A002330.

Analogs for 3 squares: A261904 and A261915.

A122923 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, sequentially minimizing the value of x, y and z. a(n) is that z.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Sep 19 2006

nonn

			10 = 2^2 + 2^2 + 1^2 + 1^2, so a(10) = 1. The only representation for 11 is 3^2 + 1^2 + 1^2 + 0^2, so a(11) = 1.

Cf. A122921, A122922, A122924.

A122924 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, sequentially minimizing the value of x, y and z. a(n) is that w.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Sep 19 2006

nonn

			10 = 2^2 + 2^2 + 1^2 + 1^2, so a(10) = 1. The only representation for 11 is 3^2 + 1^2 + 1^2 + 0^2, so a(11) = 1.

Cf. A122921, A122922, A122923, A002331.

A178786 Express n as the sum of four squares, x^2+y^2+z^2+w^2, with x>=y>=z>=w>=0, maximizing the value of x. Then a(n) is that x.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 3, 4, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 7, 8, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 9, 10

Offset: 0

Sébastien Dumortier, Jun 24 2011

nonn

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares.

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python program

Cf. A122922, A122923, A122924, A122925, A122926, A122927, A002330, A122921.

Analogs for 3 squares: A261904 and A261915.

from math import *
for nbre in range(0, 500): # or more than 500 !
    maxc4=0
    for c1 in range(0, int(sqrt(nbre/4))+1):
        for c2 in range(c1, int(sqrt(nbre/3))+1):
            for c3 in range(c2, int(sqrt(nbre/2))+1):
                s3=c3**2+c2**2+c1**2
                if s3<=nbre:
                    c4=sqrt(nbre-s3)
                    if int(c4)==c4 and c4>=c3:
                        if c4>maxc4:
                            maxc4=int(c4)
    print(maxc4, end=', ')

A307510 a(n) is the greatest product ijk*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 6, 16, 0, 12, 4, 9, 24, 8, 18, 0, 16, 36, 12, 32, 0, 24, 54, 0, 48, 20, 36, 81, 40, 72, 30, 64, 0, 60, 108, 45, 96, 40, 90, 48, 80, 144, 60, 135, 72, 120, 54, 0, 192, 108, 180, 96, 160, 72, 162, 256, 144, 240, 100

Offset: 0

Rémy Sigrist, Apr 11 2019

The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.

			For n = 34:
- 34 can be expressed in 4 ways as a sum of four squares:
    i^2 + j^2 + k^2 + l^2   i*j*k*l
    ---------------------   -------
    0^2 + 0^2 + 3^2 + 5^2         0
    0^2 + 3^2 + 3^2 + 4^2         0
    1^2 + 1^2 + 4^2 + 4^2        16
    1^2 + 2^2 + 2^2 + 5^2        20
- a(34) = max(0, 16, 20) = 20.

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of the first 20000 terms (where the color is function of the parity of n)
Rémy Sigrist, C program for A307510
Wikipedia, Lagrange's four-square theorem

See A307531 for the additive variant.

Cf. A000534, A002635, A122922.

C
```
See Links section.
```

g:= proc(n, k) option remember; local a;
  if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
  max(0,seq(a*procname(n-a^2, k-1), a=1..floor(sqrt(n))))
end proc:
seq(g(n, 4), n=0..100); # Robert Israel, Apr 15 2019

Array[Max[Times @@ # & /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)

a(n) = 0 iff n belongs to A000534.

a(n) <= (n/4)^2, with equality if and only if n is an even square. - Robert Israel, Apr 15 2019

cp's OEIS Frontend

A122921 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, minimizing the value of x. a(n) is that x.

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A122923 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, sequentially minimizing the value of x, y and z. a(n) is that z.

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A122924 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, sequentially minimizing the value of x, y and z. a(n) is that w.

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A178786 Express n as the sum of four squares, x^2+y^2+z^2+w^2, with x>=y>=z>=w>=0, maximizing the value of x. Then a(n) is that x.

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A307510 a(n) is the greatest product i*j*k*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

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A307510 a(n) is the greatest product ijk*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.