A122927 -id:A122927 - 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.

A047801 Number of different values of i^2+j^2+k^2+l^2 for i,j,k,l in [ 0,n ].

Original entry on oeis.org

1, 5, 14, 29, 50, 74, 111, 149, 197, 247, 308, 370, 451, 526, 613, 706, 815, 914, 1037, 1146, 1284, 1416, 1565, 1698, 1876, 2030, 2209, 2380, 2578, 2757, 2972, 3168, 3401, 3612, 3850, 4071, 4339, 4575, 4843, 5089, 5388

Offset: 0

Wouter Meeussen

Robin Visser, Table of n, a(n) for n = 0..5000 (terms n = 0..100 from T. D. Noe).

Cf. A034966, A047800.

Partial sums of A122927.

Table[ Length@Union@Flatten@Table[ i^2+j^2+k^2+l^2, {i, 0, n}, {j, i, n}, {k, j, n}, {l, k, n} ], {n, 0, 48} ]

Definition corrected by Jonathan Vos Post, Nov 14 2007

A122925 Index of first occurrence of n in A122921.

Original entry on oeis.org

0, 1, 5, 11, 24, 39, 53, 83, 96, 155, 176, 257, 224, 335, 376, 499, 384, 687, 701, 899, 704, 1043, 1104, 1379, 896, 1559, 1584, 1883, 1504, 2239, 2096, 2617, 1536, 2963, 2864, 3259, 2912, 3761, 3728, 3956, 2816, 4529, 4304, 5276, 4416, 5588, 5688, 5849

Offset: 0

Franklin T. Adams-Watters, Sep 19 2006

nonn

The last occurrence of n in A122921 is at 4n^2.

Cf. A122921, A122926, A122927.

A122926 Index of first occurrence of n or larger in A122921.

Original entry on oeis.org

0, 1, 5, 11, 24, 39, 53, 83, 96, 155, 176, 224, 224, 335, 376, 384, 384, 687, 701, 704, 704, 896, 896, 896, 896, 1504, 1504, 1504, 1504, 1536, 1536, 1536, 1536, 2816, 2816, 2816, 2816, 2816, 2816, 2816, 2816, 3584, 3584, 3584, 3584, 3584, 3584, 3584, 3584

Offset: 0

Franklin T. Adams-Watters, Sep 19 2006

nonn

Smallest number that cannot be represented as the sum of four squares using only numbers less than n.

Cf. A122921, A122925, A122927.

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=', ')

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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A047801 Number of different values of i^2+j^2+k^2+l^2 for i,j,k,l in [ 0,n ].

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Mathematica

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A122925 Index of first occurrence of n in A122921.

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A122926 Index of first occurrence of n or larger in A122921.

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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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Python