A000408 Numbers that are the sum of three nonzero squares.

3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29, 30, 33, 34, 35, 36, 38, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 104

Offset

1,1

Comments

a(n) !== 7 (mod 8). - Boris Putievskiy, May 05 2013

A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015

According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 nonzero squares (i.e., is in A000378 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1,2,5,10,13,25,37,58,85,130,?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017

References

L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 267.

Savin Réalis, Answer to question 25 ("Toute puissance entière de 3 est une somme de trois carrés premiers avec 3"), Mathesis 1 (1881), pp. 87-88. (See also p. 73 where the question is posed.)

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Alexander Berkovich and Will Jagy, On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2, arXiv:1101.2951 [math.NT], 2011.

B. Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), J. Int. Seq. 16 (2013) #13.6.4.

Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.

S. Mezroui, A. Azizi, and M'hammed Ziane, On a Conjecture of Farhi , J. Int. Seq. 17 (2014) #14.1.8.

Index entries for sequences related to sums of squares

Formula

a(n) = 6n/5 + O(n/sqrt(log n)). (Can the error term be improved?) - Charles R Greathouse IV, Mar 14 2014

Maple

N:= 1000: # to get all terms <= N
S:= series((JacobiTheta3(0, q)-1)^3, q, 1001):
select(t -> coeff(S, q, t)>0, [$1..N]); # Robert Israel, Jan 14 2016

Mathematica

f[n_] := Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, n}]^3, x]], n^2], _?Positive]]; f[11] (* Ray Chandler, Dec 06 2006 *)
pr[n_] := Select[ PowersRepresentations[n, 3, 2], FreeQ[#, 0] &]; Select[ Range[104], pr[#] != {} &] (* Jean-François Alcover, Apr 04 2013 *)
max = 1000; s = (EllipticTheta[3, 0, q] - 1)^3 + O[q]^(max+1); Select[ Range[max], SeriesCoefficient[s, {q, 0, #}] > 0 &] (* Jean-François Alcover, Feb 01 2016, after Robert Israel *)

Prog

(PARI) is(n)=for(x=sqrtint((n-1)\3)+1, sqrtint(n-2), for(y=1, sqrtint(n-x^2-1), if(issquare(n-x^2-y^2), return(1)))); 0 \\ Charles R Greathouse IV, Apr 04 2013
(PARI) is(n)= my(a, b) ; a=1 ; while(a^2+1<n, b=1 ; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ;
for(n=3, 1e3, if(is(n), print1(n, ", "))); \\ Altug Alkan, Jan 18 2016
(Haskell)
a000408 n = a000408_list !! (n-1)
a000408_list = filter ((> 0) . a025427) [1..]
-- Reinhard Zumkeller, Feb 26 2015
(Python)
def aupto(lim):
  squares = [k*k for k in range(1, int(lim**.5)+2) if k*k <= lim]
  sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
  sum3sqs = set(a+b for a in sum2sqs for b in squares)
  return sorted(set(range(lim+1)) & sum3sqs)
print(aupto(104)) # Michael S. Branicky, Mar 06 2021

Crossrefs

Cf. A000378, A000404, A000414, A003072, A004214, A024795, A024796, A025321, A025427.

Sequence in context: A065940 A358350 A024795 * A025321 A153238 A343112

Adjacent sequences: A000405 A000406 A000407 * A000409 A000410 A000411

Keyword

nonn

Author

N. J. A. Sloane and J. H. Conway

Status

approved