A004214 -id:A004214 - cp's OEIS frontend

A134739 Cubes of (positive numbers that are not the sum of three nonzero squares), that is, the terms of A004214, cubed.

1, 8, 64, 125, 343, 512, 1000, 2197, 3375, 4096, 8000, 12167, 15625, 21952, 29791, 32768, 50653, 59319, 64000, 103823, 140608, 166375, 195112, 216000, 250047, 262144, 357911, 493039, 512000, 614125, 658503, 778688, 857375, 1000000

Offset: 1

Artur Jasinski, Nov 07 2007

nonn

This sequence was inspired by e-mail from Ray Chandler, Nov 07 2007.
Original name was: Cubes which are not the sum of three nonzero squares. That definition would not include 125 = 5^2 + 6^2 + 8^2. - Robert Israel, Jan 12 2016
For "(cubes of positive numbers) that are not the sum of three nonzero squares", that is, the cubes in A004214, see A267189. - N. J. A. Sloane, Jan 18 2016

			8 is in the sequence because it is not possible to express 2 as a sum of three nonzero squares and 2^3 = 8.
27 is not in the sequence because 3 = 1^2 + 1^2 + 1^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004214, A134738, A267189.

N:= 1000: # to get all terms <= N^3
A004214:= {$1..N} minus {seq(seq(seq(a^2 + b^2 + c^2, c = b .. floor(sqrt(N-a^2-b^2))), b = a .. floor(sqrt(N-a^2))),a=1..floor(sqrt(N/2)))}:
map(`^`,sort(convert(A004214,list)), 3); # Robert Israel, Jan 12 2016

searchMax = 16; Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, searchMax}]^3, x]], searchMax^2], 0]]^3 (* Based on Ray Chandler's program for A004214, Alonso del Arte, Jan 12 2016 *)

is(n) = { my(a, b) ; a=1; while(a^2+1Altug Alkan, Jan 13 2016

a(n) = A004214(n)^3. - Ray Chandler, Jan 29 2009

Definition corrected by Robert Israel, Jan 12 2016

A214328 Intersection of A004214 and A018825.

Original entry on oeis.org

1, 4, 7, 15, 16, 23, 28, 31, 39, 47, 55, 60, 63, 64, 71, 79, 87, 92, 95, 103, 111, 112, 119, 124, 127, 135, 143, 151, 156, 159, 167, 175, 183, 188, 191, 199, 207, 215, 220, 223, 231, 239, 240, 247, 252, 255, 256, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343, 348, 351, 359, 367, 368, 375, 380, 383, 391, 399, 407, 412, 415, 423, 431, 439

Offset: 1

N. J. A. Sloane, Jul 26 2012, following a suggestion from Hans Isdahl, Apr 19 2012

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408, A004214, A000404, A018825, A214329.

a(n) ~ 6 * n. - Bill McEachen, Mar 24 2024

A060686 Duplicate of A004214.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 15, 16, 20, 23, 25, 28, 31, 32, 37, 39, 40, 47, 52, 55, 58, 60, 63

Offset: 1

dead

A000408 Numbers that are the sum of three nonzero squares.

Original entry on oeis.org

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

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

nonn

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

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.)

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

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

a000408 n = a000408_list !! (n-1)
a000408_list = filter ((> 0) . a025427) [1..]
-- Reinhard Zumkeller, Feb 26 2015

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

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 *)

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

is(n)= my(a, b) ; a=1 ; while(a^2+1Altug Alkan, Jan 18 2016

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

a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014; error term improved Jul 05 2024

A004215 Numbers that are the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71, 79, 87, 92, 95, 103, 111, 112, 119, 124, 127, 135, 143, 151, 156, 159, 167, 175, 183, 188, 191, 199, 207, 215, 220, 223, 231, 239, 240, 247, 252, 255, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343

Offset: 1

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

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares.
If n is in the sequence and k is an odd positive integer then n^k is in the sequence because n^k is of the form 4^i(8j+7). - Farideh Firoozbakht, Nov 23 2006
Numbers whose cubes do not have a partition as a sum of 3 squares. a(n)^3 = A134738(n). - Artur Jasinski, Nov 07 2007
A002828(a(n)) = 4; A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015
There are infinitely many adjacent pairs (for example, 128n + 111 and 128n + 112 for any n), but never a triple of consecutive integers. - Ivan Neretin, Aug 17 2017
These numbers are called "forbidden numbers" in crystallography: for a cubic crystal, no reflection with index hkl such that h^2 + k^2 + l^2 = a(n) appears in the crystal's diffraction pattern. - A. Timothy Royappa, Aug 11 2021

			15 is in the sequence because it is the sum of four squares, namely, 3^2 + 2^2 + 1^2 + 1^2, and it can't be expressed as the sum of fewer squares.
16 is not in the sequence, because, although it can be expressed as 2^2 + 2^2 + 2^2 + 2^2, it can also be expressed as 4^2.

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. 261.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 12.
E. Poznanski, 1901. Pierwiastki pierwotne liczb pierwszych. Warszawa, pp. 1-63.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p. 125).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 4181.

T. D. Noe, Table of n, a(n) for n = 1..10000
David S. Bettes, Letter to N. J. A. Sloane, Nov 05 1976
Richard T. Bumby, Sums Of Four Squares
International Union of Crystallography, Cubic structures.
Shuo Li, The characteristic sequence of the integers that are the sum of two squares is not morphic, arXiv:2404.08822 [math.NT], 2024.
Louis J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
Saburô Uchiyama, A five-square theorem, Publ. Res. Math. Sci., Vol 13, Number 1 (1977), 301-305.
Steve Waterman, Missing numbers formula
Eric Weisstein's World of Mathematics, Square Number
Wikipedia, Lagrange's four-square theorem.
Index entries for sequences related to sums of squares

Complement of A000378.
Cf. A002828, A055039, A072401, A125084, A134738, A134739, A055045, A055046, A234000.
Cf. A000118 (ways to write n as sum of 4 squares), A025427.

a004215 n = a004215_list !! (n-1)
a004215_list = filter ((== 4) . a002828) [1..]
-- Reinhard Zumkeller, Feb 26 2015

N:= 1000: # to get all terms <= N
{seq(seq(4^i * (8*j + 7), j = 0 .. floor((N/4^i - 7)/8)), i = 0 .. floor(log[4](N)))}; # Robert Israel, Sep 02 2014

Sort[Flatten[Table[4^i(8j + 7), {i, 0, 2}, {j, 0, 42}]]] (* Alonso del Arte, Jul 05 2005 *)
Select[Range[120], Mod[ #/4^IntegerExponent[ #, 4], 8] == 7 &] (* Ant King, Oct 14 2010 *)

isA004215(n)={ local(fouri,j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1,400, if(isA004215(n), print1(n,",") ; ) ; ) ; } \\ R. J. Mathar, Nov 22 2006

isA004215(n)= n\4^valuation(n,4)%8==7 \\ M. F. Hasler, Mar 18 2011

def valuation(n, b):
    v = 0
    while n > 1 and n%b == 0: n //= b; v += 1
    return v
def ok(n): return n//4**valuation(n, 4)%8 == 7 # after M. F. Hasler
print(list(filter(ok, range(344)))) # Michael S. Branicky, Jul 15 2021

from itertools import count, islice
def A004215_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
A004215_list = list(islice(A004215_gen(),30)) # Chai Wah Wu, Jul 09 2022

def A004215(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>(i<<1))-7>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

a(n) = A055039(n)/2. - Ray Chandler, Jan 30 2009
Numbers of the form 4^i*(8*j+7), i >= 0, j >= 0. [A.-M. Legendre & C. F. Gauss]
Products of the form A000302(i)*A004771(j), i, j >= 0. - R. J. Mathar, Nov 29 2006
a(n) = 6*n + O(log(n)). - Charles R Greathouse IV, Dec 19 2013
Conjecture: The number of terms < 2^n is A023105(n) - 2. - Tilman Neumann, Sep 20 2020

More terms from Arlin Anderson (starship1(AT)gmail.com)

Additional comments from Jud McCranie, Mar 19 2000

A025427 Number of partitions of n into 3 nonzero squares.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 2, 0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 3, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 1, 3, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 2, 1, 0, 0, 3, 1, 2, 1, 0, 3, 0, 1, 3, 2, 1, 0, 1, 2, 0, 1, 1, 2, 3, 0, 3, 2, 0, 1, 2, 1, 2

Offset: 0

David W. Wilson

The non-vanishing values a(n) give the multiplicities for the numbers n appearing in A000408. See also A024795 where these numbers n are listed a(n) times. For the primitive case see A223730 and A223731. - Wolfdieter Lang, Apr 03 2013

			a(27) = 2 because  1^2 + 1^2 + 5^2 = 27  = 3^2 + 3^2 + 3^2. The second representation is not primitive (gcd(3,3,3) = 3 not 1).

R. J. Mathar and R. Zumkeller, Table of n, a(n) for n = 0..10000, first 5592 terms from R. J. Mathar
Index to sequences related to sums of squares and cubes.

Cf. A000408, A024795, A223730 (multiplicities for the primitive case). - Wolfdieter Lang, Apr 03 2013
Column k=3 of A243148.
Cf. A000290, A010052, A004214, A025321, A025414, A025426.

a025427 n = sum $ map f zs where
   f x = sum $ map (a010052 . (n - x -)) $
                   takeWhile (<= div (n - x) 2) $ dropWhile (< x) zs
   zs = takeWhile (< n) $ tail a000290_list
-- Reinhard Zumkeller, Feb 26 2015

A025427 := proc(n)
    local a,x,y,zsq ;
    a := 0 ;
    for x from 1 do
        if 3*x^2 > n then
            return a;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            end if;
            zsq := n-x^2-y^2 ;
            if issqr(zsq) then
                a := a+1 ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 3):
seq(a(n), n=0..107);  # Alois P. Heinz, Jun 14 2025

Count[PowersRepresentations[#, 3, 2], pr_ /; (Times @@ pr) > 0]& /@ Range[0, 120] (* Jean-François Alcover, Jan 30 2018 *)

a(n)=if(n<3, return(0)); sum(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); sum(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), issquare(t-j^2))) \\ Charles R Greathouse IV, Aug 05 2024

a(A004214(n)) = 0; a(A000408(n)) > 0; a(A025414(n)) = n and a(m) != n for m < A025414(n). - Reinhard Zumkeller, Feb 26 2015
a(4n) = a(n). This is because if a number divisible by 4 is the sum of three squares, each of those squares must be even. - Robert Israel, Mar 09 2016
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010052(i) * A010052(k) * A010052(n-i-k). - Wesley Ivan Hurt, Apr 19 2019
a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

A051952 Numbers that are not a sum of 3 positive squares nor are of the form 4^a*(8b+7) and which are not multiples of 4.

Original entry on oeis.org

1, 2, 5, 10, 13, 25, 37, 58, 85, 130

Offset: 1

Eberhard R. Hilf, Dec 21 1999

The asymptotic eigenvalue spectrum of the Schroedinger equation for a free particle in a box in three dimensions is known only (that is: average level density and average degeneracy) if the a(n) are finite series.
It is not known whether 130 is the largest such number or if this is the start of an infinite series.
See Theorem 6 of Grosswald's book, p. 79: every positive integer n not of the form 4^a*(8*m+7), with a and m nonnegative integers [i.e., n is a sum of three squares, proved by Legendre (1798)] and not of the form 4^b*a(n), with b a nonnegative integer, n=1,...,10, and possibly one a(11) > 5*10^10, is a sum of three non-vanishing squares. See A004214 and A000408. In the F. Halter-Koch reference, p. 13, one finds a Korollar 1 (b) stating for positive integers n, not 0, 4, 7 modulo 8: n is not the sum of three positive coprime squares if and only if n = a(n), n=1,...,10, and possibly one more number a(11) >= 5*10^10. See A223731. - Wolfdieter Lang, Apr 04 2013

			Consider a(3)=5: 1^2 +1^2 +1^2=3, too low; 1^2+1^2+2^2=6, too high. 4^1=4 too low; 4^2=16 too high; (8*0+7)=7 too low, (8*1+7)= 15 too high; thus 5 is a member of this sequence.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 79 with p. 76.

H. P. Baltes and E. R. Hilf, Spectra of finite systems; BI-Verlag.
H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
H.-P. Baltes, Peter K. J. Draxl and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés strictement positifs, Mémoires de la Société Mathématique de France, 37 (1974), p. 53-53.
E. Grosswald, A. Calloway and J. Calloway, The representations of integers by three positive squares, Proc. Amer. Math. Soc. 10 (1959), 451-455. [Math. Rev. 21 #3376; E24-73 in Leveque's Reviews in Number Theory, Vol. 2, p. 290]
F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, p. 13.
Eberhard R. Hilf, Publications
Eberhard R. Hilf, Über den Oberflächenterm der Gesamtenergie der Atomkerne nach dem Fermigas-Modell, Diploma-thesis, Universität Frankfurt, Germany, 1963.
E. R. Hilf and H. P. Baltes, 130 and the cube spectrum, unpublished
E. R. Hilf, G. Suessmann, Surface Tension of nuclei according to the Fermi-gas-model, Physics Letters, Vol. 21, No. 6, p. 654-656, (1966).
Index entries for sequences related to sums of squares

nmax = 1000; amax = Ceiling[Log[nmax/7]/Log[4]]; notThreeSquaresQ[n_] := Reduce[0 < a <= b <= c && n == a^2 + b^2 + c^2, {a, b, c}, Integers] === False; notOfTheFormQ[n_, a_] := Reduce[n == 4^a*(8*b+7), b, Integers] === False; notOfTheFormQ[n_] := And @@ (notOfTheFormQ[n, #] & ) /@ Range[0, amax]; Select[Range[nmax], !Mod[#, 4] == 0 && notThreeSquaresQ[#] && notOfTheFormQ[#] & ](* Jean-François Alcover, Jun 12 2012 *)

Grosswald et al. reference from N. J. A. Sloane, Jun 07 2000

A135693 Not the sum of three positive squares or cubes.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 15, 23, 31, 87, 111, 119, 148, 167, 263, 311, 335, 391, 407, 455, 559, 599, 839, 951, 1159, 1231, 1287, 1303, 1391, 1455, 1463, 1607, 1660, 1679, 1751, 1863, 1991, 2351, 2615, 2799, 3247, 3983, 4327, 4367, 5199, 5655, 6047, 6159, 6351

Offset: 1

Zak Seidov and Giovanni Resta, Feb 24 2008

nonn

No other terms < 10^9. Presumably the sequence is finite.

Sum can include a mix of squares and cubes. - James C. McMahon, Apr 19 2025

Chai Wah Wu, Table of n, a(n) for n = 1..82

Cf. A113505, A135367.

Cf. A004214, A057904.

lim=40000; s=Range[Sqrt[lim]]^2; c=Range[Surd[lim, 3]]^3; A135693=Complement[Range[lim], Select[Total/@Tuples[Union[s, c], {3}], #<=lim&]] (* James C. McMahon, Apr 19 2025 *)

Edited by N. J. A. Sloane, Mar 01 2008

A000549 Numbers that are the sum of 2 squares but not sum of 3 nonzero squares.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 16, 20, 25, 32, 37, 40, 52, 58, 64, 80, 85, 100, 128, 130, 148, 160, 208, 232, 256, 320, 340, 400, 512, 520, 592, 640, 832, 928, 1024, 1280, 1360, 1600, 2048, 2080, 2368, 2560, 3328, 3712, 4096, 5120, 5440, 6400, 8192, 8320, 9472, 10240

Offset: 1

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

nonn

It appears that starting at 64, the sequence follows the pattern, 2^(2n) + {0, 16, 21, 36, 64, 66, 84, 96, 144, 168} * 2^(2n-6) for n >= 3. - T. D. Noe, Jun 20 2012

Intersection of A001481 and A004214. - Ralf Stephan, Mar 28 2014

Robert Israel, Table of n, a(n) for n = 1..113 (first 100 terms from T. D. Noe)
Index entries for sequences related to sums of squares

Asq = zeros(1,10^7);
Asq( [0:3162] .^ 2 + 1) = 1;
Psq = zeros(1,10^7);
Psq( [1:3162] .^ 2 + 1) = 1;
As2 = conv(Asq,Asq);
As2 = min(1, As2(1:10^7));
Ps2 = conv(Psq,Psq);
Ps2 = min(1, Ps2(1:10^7));
As3 = conv(Asq,As2);
As3 = min(1, As3(1:10^7));
Ps3 = conv(Psq,Ps2);
Ps3 = min(1, Ps3(1:10^7));
D = As3 - Ps3;
A000549 = find(D(2:end))
% Robert Israel, Mar 26 2014

# this requires Maple 17 or later
  m:= 5000: # to find all entries <= m^2
N:= m^2:
with(SignalProcessing):
with(ArrayTools):
A1:= Array(0..N,datatype=float[8]):
for i from 1 to m do A1[i^2]:= 1 end do:
B:= Convolution(A1,A1):
A2:= Array(0..N,datatype=float[8]):
Copy(N+1,B,A2):
All1:= Array(0..N,datatype=float[8],fill=1):
MinimumEvery(A2,All1,container=A2):
B:= Convolution(A1,A2):
A3:= Array(0..N,datatype=float[8]):
Copy(N+1,B,A3);
MinimumEvery(A3,All1,container=A3);
MaximumEvery(A1,A2,container=A2);
B:= evalhf(map(round,A3-A2));
MinimumEvery(B,Array(0..N,datatype=float[8]),container=B);
R:= ArrayTools[SearchArray](B);
A000549:= map(`-`,convert(R,list),1); # Robert Israel, Mar 28 2014

A000549 = Reap[For[n=1, n <= 10^5, n++, If[SquaresR[2, n] != 0, If[Union[ PowersRepresentations[n, 3, 2][[All, 1]]] == {0}, Print[n]; Sow[n]]]] ][[2, 1]] (* Jean-François Alcover, Feb 09 2016 *)

istwo(n)=if(n<3,return(n>=0)); my(f=factor(n>>valuation(n,2))); for(i=1,#f~, if(f[i,2]%2&&f[i,1]%4==3, return(0))); 1
is(n)=if(!istwo(n),return(0)); if(n<3, return(1)); for(x=sqrtint(n\3),sqrtint(n-2), my(t=n-x^2); for(y=sqrtint(t\2),sqrtint(t-1), if(t&&issquare(t-y^2), return(0)))); 1 \\ Charles R Greathouse IV, Jan 28 2016

Empirical g.f.: x*(1 +2*x +4*x^2 +5*x^3 +8*x^4 +10*x^5 +13*x^6 +16*x^7 +20*x^8 +25*x^9 +28*x^10 +29*x^11 +24*x^12 +32*x^13 +26*x^14 +24*x^15 +28*x^16 +21*x^17 +20*x^18 +28*x^19 +2*x^20) / ((1 -2*x^5)*(1 +2*x^5)). - Colin Barker, Feb 09 2016

Extended by T. D. Noe, Jun 20 2012

A123120 Numbers k>0 such that m+k is not the sum of m nonzero squares for any m>5.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 13

Offset: 1

Alexander Adamchuk, Sep 28 2006

Also, numbers which are not the sum of "squares-minus-1" (cf. A005563). - Benoit Jubin, Apr 14 2010
Conjecture: All but (n+6) positive numbers are equal to the sum of n>5 nonzero squares. For all n>5 the only (n+6) positive numbers that are not equal to the sum of n nonzero squares are {1,2,3,...,n-3,n-2,n-1,n+1,n+2,n+4,n+5,n+7,n+10,n+13}.
  Numbers that are not squares (n=1): A000037 = {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28,...}
  Numbers that are not the sum of 2 nonzero squares (n=2): A018825 = {1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 16, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 36,...}.
  Numbers that are not the sum of three nonzero squares (n=3): A004214 = {1, 2, 4, 5, 7, 8, 10, 13, 15, 16, 20, 23, 25, 28, 31, 32, 37, 39, 40, 47, 52, 55, 58,...}.
  Numbers that are not the sum of 4 nonzero squares (n=4): A000534 ={1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512,...}.
  Numbers that are not the sum of 5 nonzero squares (n=5): A047701 = {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33}.
  Numbers that are not the sum of 6 nonzero squares (n=6): {1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19}.
  Numbers that are not the sum of 7 nonzero squares (n=7): {1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20}.
  Numbers that are not the sum of 8 nonzero squares (n=8): {1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 18, 21}.
  Numbers that are not the sum of 9 nonzero squares (n=9): {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 19, 22}.
The above conjecture appears as Theorem 2 on p. 73 in the Grosswald reference, where it is attributed to E. Dubouis (1911). - Wolfdieter Lang, Mar 27 2013

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, pp. 73-74.

Cf. A000037, A018825, A004214, A000534, A047701.

Edited by M. F. Hasler, Feb 23 2018

cp's OEIS Frontend

A134739 Cubes of (positive numbers that are not the sum of three nonzero squares), that is, the terms of A004214, cubed.

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A214328 Intersection of A004214 and A018825.

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A060686 Duplicate of A004214.

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A000408 Numbers that are the sum of three nonzero squares.

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A004215 Numbers that are the sum of 4 but no fewer nonzero squares.

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A025427 Number of partitions of n into 3 nonzero squares.

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A051952 Numbers that are not a sum of 3 positive squares nor are of the form 4^a*(8b+7) and which are not multiples of 4.

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