A261904 -id:A261904 - 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.

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=', ')

A261915 Smallest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 11 2015

sign

a(n) = -1 iff n is in A004215, a(n) >= 0 iff n is in A000378.

If we maximize x we get A261904.

			Table showing initial values of n,x,y,z:
   0  0  0  0
   1  1  0  0
   2  1  1  0
   3  1  1  1
   4  2  0  0
   5  2  1  0
   6  2  1  1
   7 -1 -1 -1
   8  2  2  0
   9  2  2  1
  10  3  1  0
  11  3  1  1
  12  2  2  2
  13  3  2  0
  14  3  2  1
  15 -1 -1 -1
  16  4  0  0
  17  3  2  2
  18  3  3  0
  19  3  3  1
  20  4  2  0
  ...

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program
Index entries for sequences related to sums of squares

Cf. A000378, A004215, A005875, A261904.

Analogs for 4 squares: A178786 and A122921.

a(17) corrected, more terms from David Consiglio, Jr., Sep 11 2015

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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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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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A261915 Smallest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

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