A000534 Numbers that are not the sum of 4 nonzero squares.

0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896, 1536, 2048, 3584, 6144, 8192, 14336, 24576, 32768, 57344, 98304, 131072, 229376, 393216, 524288, 917504, 1572864, 2097152, 3670016, 6291456, 8388608, 14680064

Offset

1,3

Comments

For n > 15, a(n) = A006431(n-1). - Thomas Ordowski, Nov 18 2012

References

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 302.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, Theorem 3, pp. 74-75.

Links

T. D. Noe, Table of n, a(n) for n = 1..100

Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. See p. 2.

Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.

Index entries for linear recurrences with constant coefficients, signature (0,0,4).

Index entries for sequences related to sums of squares

Formula

Consists of the numbers 0, 1, 3, 5, 9, 11, 17, 29, 41, 2*4^m, 6*4^m and 14*4^m (m >= 0). Compare A123069.

From 224 on, a(n) = 4*a(n-3).

Numbers n such that A025428(n) = 0.

G.f.: x^2*(36*x^16 + 32*x^15 + 60*x^14 + 55*x^13 + 36*x^12 + 27*x^11 + 20*x^10 + 19*x^9 + 18*x^8 + 13*x^7 + 11*x^6 + 4*x^5 + 2*x^4 - x^3 - 3*x^2 - 2*x - 1)/(4*x^3 - 1). - Chai Wah Wu, Jul 09 2022

Mathematica

q=22; lst={}; Do[Do[Do[Do[z=a^2+b^2+c^2+d^2; If[z<=q^2+3, AppendTo[lst, z]], {d, q}], {c, q}], {b, q}], {a, q}]; lst1=Union@lst lst={}; Do[AppendTo[lst, n], {n, q^2+3}]; lst2=lst Complement[lst2, lst1] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Join[{0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41}, LinearRecurrence[{0, 0, 4}, {56, 96, 128}, 30]] (* Jean-François Alcover, Feb 09 2016 *)

Prog

(PARI) for(n=1, 224, if(sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, if(a^2+b^2+c^2+d^2-n, 0, 1)))))==0, print1(n, ", ")))
(PARI) {a(n)=if( n<2, 0, n<16, [1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41][n-1], [4, 7, 12][n%3+1] * 2^(n\3*2-7))}; /* Michael Somos, Apr 23 2006 */
(PARI) is(n)=my(k=if(n, n/4^valuation(n, 4), 2)); k==2 || k==6 || k==14 || setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014
(Python)
from itertools import count, islice
def A000534_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2, 6, 14}, count(max(startvalue, 0)))
A000534_list = list(islice(A000534_gen(), 30)) # Chai Wah Wu, Jul 09 2022

Crossrefs

Cf. A123069, A000414 (complement).

Sequence in context: A027563 A219729 A335659 * A136112 A127936 A280771

Adjacent sequences: A000531 A000532 A000533 * A000535 A000536 A000537

Keyword

nonn,easy,nice,changed

Author

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

Status

approved