A005767 Solutions k to k^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

1,1

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

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

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

For n>=32, a(n) = n+m+1 if n+m>=A094958(m+1) and a(n) = n+m otherwise, where m = A070939(n)+A070939(floor(n/5)). - Chai Wah Wu, Jan 25 2026

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 *)

Prog

(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
(Python)
def A005767(n): return (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)[n-1] if n<32 else n+(m:=n.bit_length()+(n//5).bit_length())+(n+m>=(1<<m if m<3 else 5<<((m+1>>1)-2) if m&1 else 1<<(m+1>>1)+1)) # Chai Wah Wu, Jan 25 2026

Crossrefs

Complement of A094958. Cf. A169580, A000378, A000419, A000408, A070939.

For primitive solutions see A005818.

Sequence in context: A288938 A282140 A071530 * A085837 A176237 A187811

Adjacent sequences: A005764 A005765 A005766 * A005768 A005769 A005770

Keyword

nonn,easy

Author

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

Extensions

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

Status

approved