A285552 Smallest number k that cannot be expressed as x^2 + y^2 + z^2 + w^2 where x >= y >= z >= w >= 0 and x > floor(sqrt(k)) - n, but can be so expressed if x = floor(sqrt(k)) - n.

23, 224, 128, 3712, 896, 512, 1536, 54272, 14848, 11264, 3584, 11776, 2048, 6144, 20480, 833536, 217088, 94208, 59392, 45056, 116736, 22528, 14336, 118784, 47104, 8192, 63488, 24576, 49152, 81920, 294912, 13082624, 3334144, 1564672, 868352, 548864, 376832

Offset: 1

Author

Jon E. Schoenfield, Jun 14 2017

Keywords

• nonn

Comments

Lagrange's theorem tells us that each positive integer k can be written as a sum of four squares. Some can be written as such a sum using a "greedy" algorithm in which x = floor(sqrt(k)), y = floor(sqrt(k - x^2)), z = floor(sqrt(k - x^2 - y^2)), and w = sqrt(k - x^2 - y^2 - z^2); e.g., 165 = 12^2 + 4^2 + 2^2 + 1^2, and x = floor(sqrt(165)) = floor(12.845...) = 12. For some other positive integers k, there is no sum x^2 + y^2 + z^2 + w^2 = k in which x = floor(sqrt(k)), no matter what values of y, z, and w are tested. For certain positive integers k, the largest x such that x^2 + y^2 + z^2 + w^2 = k is considerably less than floor(sqrt(k)). a(n) is the smallest number k such that there is no such sum in which floor(sqrt(k)) - x < n, but there is at least one such sum in which floor(sqrt(k)) - x = n.

Larger terms tend to be divisible by larger powers of two:

n a(n)

== ==================

1 23 = 2^0 * 23

2 224 = 2^5 * 7

3 128 = 2^7 * 1

4 3712 = 2^7 * 29

5 896 = 2^7 * 7

6 512 = 2^9 * 1

7 1536 = 2^9 * 3

8 54272 = 2^10 * 53

9 14848 = 2^9 * 29

10 11264 = 2^10 * 11

11 3584 = 2^9 * 7

12 11776 = 2^9 * 23

13 2048 = 2^11 * 1

14 6144 = 2^11 * 3

15 20480 = 2^12 * 5

16 833536 = 2^11 * 407

17 217088 = 2^12 * 53

18 94208 = 2^12 * 23

19 59392 = 2^11 * 29

20 45056 = 2^12 * 11

21 116736 = 2^11 * 57

22 22528 = 2^11 * 11

23 14336 = 2^11 * 7

24 118784 = 2^12 * 29

25 47104 = 2^11 * 23

26 8192 = 2^13 * 1

27 63488 = 2^11 * 31

28 24576 = 2^13 * 3

29 49152 = 2^14 * 3

30 81920 = 2^14 * 5

31 294912 = 2^15 * 9

In some regions, a plot of the sequence looks fairly chaotic, but at each value of n = 2^k, a(n) reaches a local maximum, and the lengths of the monotonic runs on either side increases as k increases; e.g.,

a(1) < a(2) > a(3)

a(3) < a(4) > ... > a(6)

a(6) < ... < a(8) > ... > a(11)

a(13) < ... < a(16) > ... > a(20)

a(28) < ... < a(32) > ... > a(40)

a(59) < ... < a(64) > ... > a(75)

a(122) < ... < a(128) > ... > a(143)

a(248) < ... < a(256) > ... > a(275)

Examples

			At k = 128, floor(sqrt(k)) = 11, and there is no sum x^2 + y^2 + z^2 + w^2 = k in which x = 11, 10, or 9, but there is such a sum in which x = 8 (namely, 8^2 + 8^2 + 0^2 + 0^2 = 128); no smaller positive integer k has this property, so a(3) = 128.
At k = 13082624, floor(sqrt(k)) = 3616, and there is no sum x^2 + y^2 + z^2 + w^2 = k such that x > 3584 = 3616 - 32, but there is such a sum in which x = 3584 (namely, 3584^2 + 448^2 + 192^2 + 0^2 = 13082624); no smaller positive integer k has this property, so a(32) = 13082624.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A178786.