A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

Author

N. J. A. Sloane, Ralph Peterson (ralphp(AT)library.nrl.navy.mil)

Keywords

• nonn
• easy

Comments

All numbers not equal to some 2^k or 5*2^k [Fraser and Gordon]. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006

References

• T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

Crossrefs

Complement of A094958. Cf. A169580, A000378, A000419, A000408.

For primitive solutions see A005818.

Programs

Mathematica

z=100;lst={};Do[a2=a^2;Do[b2=b^2;Do[c2=c^2;e2=a2+b2+c2;e=Sqrt[e2];If[IntegerQ[e]&&e<=z,AppendTo[lst,e]],{c,b,1,-1}],{b,a,1,-1}],{a,1,z}];Union@lst (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)

PARI

is(n)=if(n%5,n,n/5)==2^valuation(n,2) \\ Charles R Greathouse IV, Mar 12 2013

Python

def A005767(n):
    def f(x): return n+x.bit_length()+(x//5).bit_length()
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

Formula

a(n) = n + 2*log_2(n) + O(1). - Charles R Greathouse IV, Sep 01 2015

A169580(n) = a(n)^2. - R. J. Mathar, Aug 15 2023

Extensions

More terms from T. D. Noe, Mar 04 2010