login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000408 Numbers that are the sum of three nonzero squares. 59
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Jonathan Vos Post, Mar 03 2010: (Start)

Catalan conjectured that three times any odd square not divisible by 5 is a sum of squares of three primes other than 2 and 3 (regarding 1 as a prime). Catalan stated and Realis proved that every power of 3 is a sum of three squares relatively prime to 3. [See Dickson]

From the abstract of the Berkovich and Jagy paper: "Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n) - ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. To prove this identity we employ the Siegel-Weil and the Smith-Minkowski product formulas."

(End)

a(n) not equal 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.)

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

LINKS

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

Alexander Berkovich, 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.

Index entries for sequences related to sums of squares

FORMULA

a(n) = 6n/5 + O(x/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

CROSSREFS

Cf. A004214 (complement), A024795 (numbers with multiplicity), A000404, A000378, A025427.

Sequence in context: A224012 A065940 A024795 * A025321 A153238 A230193

Adjacent sequences:  A000405 A000406 A000407 * A000409 A000410 A000411

KEYWORD

nonn

AUTHOR

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 23 06:43 EDT 2017. Contains 289686 sequences.