A001974 Numbers that are the sum of 3 distinct squares, i.e., numbers of the form x^2 + y^2 + z^2 with 0 <= x < y < z.

5, 10, 13, 14, 17, 20, 21, 25, 26, 29, 30, 34, 35, 37, 38, 40, 41, 42, 45, 46, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 65, 66, 68, 69, 70, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 93, 94, 97, 98, 100, 101, 104, 105, 106, 107, 109, 110, 113

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Also: Numbers which are the sum of two or three distinct nonzero squares. - M. F. Hasler, Feb 03 2013

According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 distinct squares (i.e., is in A001974 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627, ?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017

Examples

			5 = 0^2 + 1^2 + 2^2.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A001983, A024803, A025339, A025442, A004432.

Cf. A004436 (complement).

Programs

Mathematica

r[n_] := Reduce[0 <= x < y < z && x^2 + y^2 + z^2 == n, {x, y, z}, Integers]; ok[n_] := r[n] =!= False; Select[ Range[113], ok] (* Jean-François Alcover, Dec 05 2011 *)

Python

from itertools import combinations
def aupto(lim):
  s = filter(lambda x: x <= lim, (i*i for i in range(int(lim**.5)+2)))
  s3 = set(filter(lambda x: x<=lim, (sum(c) for c in combinations(s, 3))))
  return sorted(s3)
print(aupto(113)) # Michael S. Branicky, May 10 2021