A000534 -id:A000534 - cp's OEIS frontend

A000414 Numbers that are the sum of 4 nonzero squares.

4, 7, 10, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

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

As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1608 is in the sequence as 1608 = 18^2 + 20^2 + 20^2 + 22^2.
2140 is in the sequence as 2140 = 21^2 + 21^2 + 23^2 + 27^2.
3298 is in the sequence as 3298 = 25^2 + 26^2 + 29^2 + 34^2. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences related to sums of squares

Cf. A000534 (complement).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

q=16;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}];Union@lst (*Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Total/@Tuples[Range[10]^2,4]//Union (* Harvey P. Dale, Mar 18 2025 *)

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

limit = 10026 # 10000th term in b-file
from functools import lru_cache
nzs = [k*k for k in range(1, int(limit**.5)+2) if k*k + 3 <= limit]
nzss = set(nzs)
@lru_cache(maxsize=None)
def ok(n, m): return n in nzss if m == 1 else any(ok(n-s, m-1) for s in nzs)
print([n for n in range(4, limit+1) if ok(n, 4)]) # Michael S. Branicky, Apr 07 2021

from itertools import count, islice
def A000414_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not(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)))
A000414_list = list(islice(A000414_gen(),30)) # Chai Wah Wu, Jul 09 2022

a(n) = n + O(log n). - Charles R Greathouse IV, Sep 03 2014

corrected 6/95

A025428 Number of partitions of n into 4 nonzero squares.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 4, 1, 3, 2, 1, 4, 1, 1, 3, 3, 2, 2, 4, 2, 1, 3, 2, 3, 4, 2, 3, 3, 1, 2, 5, 2, 4, 3, 2, 4, 1, 1, 6, 4, 3, 4, 2, 3, 0, 4, 4, 3, 5, 1, 5, 5, 1, 4, 5, 2

Offset: 0

David W. Wilson

Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015

Cf. A000414, A000534, A025357-A025375, A216374, A025416 (greedy inverse).

Column k=4 of A243148.

A025428 := proc(n)
    local a,i,j,k,lsq ;
    a := 0 ;
    for i from 1 do
        if 4*i^2 > n then
            return a;
        end if;
        for j from i do
            if i^2+3*j^2 > n then
                break;
            end if;
            for k from j do
                if i^2+j^2+2*k^2 > n then
                    break;
                end if;
                lsq := n-i^2-j^2-k^2 ;
                if lsq >= k^2 and issqr(lsq) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc:
seq(A025428(n),n=1..40) ; # R. J. Mathar, Jun 15 2018
# 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), 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 14 2019

nn = 100; lim = Sqrt[nn]; t = Table[0, {nn}]; Do[n = a^2 + b^2 + c^2 + d^2; If[n <= nn, t[[n]]++], {a, lim}, {b, a, lim}, {c, b, lim}, {d, c, lim}]; t (* T. D. Noe, Sep 28 2012 *)
f[n_] := Length@ IntegerPartitions[n, {4}, Range[ Floor[ Sqrt[n - 1]]]^2]; Array[f, 105] (* Robert G. Wilson v, Sep 28 2012 *)

A025428(n)=sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,a^2+b^2+c^2+d^2==n))))

A025428(n)=sum(a=1,sqrtint(max(n-3,0)), sum(b=1,min(sqrtint(n-a^2-2),a), sum(c=1,min(sqrtint(n-a^2-b^2-1),b),issquare(n-a^2-b^2-c^2,&d) & d <= c )))

A025428(n)=sum(a=sqrtint(max(n,4)\4),sqrtint(max(n-3,0)), sum(b=sqrtint((n-a^2)\3-1)+1,min(sqrtint(n-a^2-2),a), sum(c=sqrtint((t=n-a^2-b^2)\2-1)+1, min(sqrtint(t-1),b), issquare(t-c^2) ))) \\ - M. F. Hasler, Sep 17 2012
for(n=1,100,print1(A025428(n),","))

T(n)={a=matrix(n,4,i,j,0);for(d=1,sqrtint(n),forstep(i=n,d*d+1,-1,for(j=2,4,a[i,j]+=sum(k=1,j,if(k0,a[i-k*d*d,j-k],if(k==j&&i-k*d*d==0,1)))));a[d*d,1]=1);for(i=1,n,print(i" "a[i,4]))} /* Robert Gerbicz, Sep 28 2012 */

For n>0, a(n) = ( A063730(n) + 6*A213024(n) + 3*A063725(n/2) + 8*A092573(n) + 6*A010052(n/4) ) / 24. - Max Alekseyev, Sep 30 2012
a(n) = ( A000118(n) - 4*A005875(n) - 6*A004018(n) - 12*A000122(n) - 15*A000007(n) + 12*A014455(n) - 24*A033715(n) - 12*A000122(n/2) + 12*A004018(n/2) + 32*A033716(n) - 32*A000122(n/3) + 48*A000122(n/4) ) / 384. - Max Alekseyev, Sep 30 2012
a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(n-i-j-k). - Wesley Ivan Hurt, Apr 19 2019

Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012

A003334 Numbers that are the sum of 11 positive cubes.

Original entry on oeis.org

11, 18, 25, 32, 37, 39, 44, 46, 51, 53, 58, 60, 63, 65, 67, 70, 72, 74, 77, 79, 81, 84, 86, 88, 89, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112, 114, 115, 116, 117, 119, 121, 122, 123, 124, 126, 128, 129, 130, 131, 133, 135, 136, 137, 138, 140, 141, 142, 143, 144

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

The sequence contains all integers greater than 321 which is the last of only 92 positive integers not in this sequence. - M. F. Hasler, Aug 25 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1120 is in the sequence as 1120 = 2^3 + 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 5^3 +  8^3.
2339 is in the sequence as 2339 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 9^3 +  9^3.
3594 is in the sequence as 3594 = 4^3 + 5^3 + 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 7^3 + 7^3 + 8^3 + 10^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Other sequences S(k, m) of numbers that are the sum of k nonzero m-th powers:
  A000404 = S(2, 2), A000408 = S(3, 2), A000414 = S(4, 2) complement of A000534,
  A047700 = S(5, 2) complement of A047701, A180968 = complement of S(6,2);
  A003325 = S(2, 3), A003072 = S(3, 3), A003327 .. A003335 = S(4 .. 12, 3) and A332107 .. A332111 = complement of S(7 .. 11, 3);
  A003336 .. A003346 = S(2 .. 12, 4), A003347 .. A003357 = S(2 .. 12, 5),
  A003358 .. A003368 = S(2 .. 12, 6), A003369 .. A003379 = S(2 .. 12, 7),
  A003380 .. A003390 = S(2 .. 12, 8), A003391 .. A004801 = S(2 .. 12, 9),
  A004802 .. A004812 = S(2 .. 12, 10), A004813 .. A004823 = S(2 .. 12, 11).

(A003334_upto(N, k=11, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ See also A003333 for alternate code. - M. F. Hasler, Aug 03 2020

a(n) = n + 92 for all n > 229. - M. F. Hasler, Aug 25 2020

A047701 All positive numbers that are not the sum of 5 nonzero squares.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

			4 = 1^2 + 1^2 + 1^2 + 1^2 + 0^2, but the last square is 0, and hence 4 is in the sequence.
5 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2, and therefore 5 is not in the sequence.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, Theorem 2., p. 73.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 145

Index entries for sequences related to sums of squares

Cf. A047700, A000534, A180968 (not the sum of 6 squares).

Select[ Range[100], Select[ PowersRepresentations[#, 5, 2], FreeQ[#, 0]& ] == {}& ] (* Jean-François Alcover, Sep 03 2013 *)

Name changed. - Wolfdieter Lang, Mar 28 2013

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

A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

Original entry on oeis.org

1, 3, 3, 1, 2, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3

Offset: 1

Yifan Xie, Apr 05 2023

a(n) is the smallest number k such that n*k can be expressed as the sum of k nonzero squares.

			For n = 2, if k = 1, 2*1 = 2 is a nonsquare; if k = 2, 2*2 = 4 cannot be expressed as the sum of 2 nonzero squares; if k = 3, 2*3 = 6 = 2^2+1^2+1^2, so a(2) = 3.

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

Yifan Xie, Table of n, a(n) for n = 1..10000
Wikipedia, Lagrange's four-square theorem
Index entries for sequences related to sums of squares

Cf. A000290, A000378, A000404, A000408, A000414, A000534, A047700.

Cf. A362068 (allows zeros), A362110 (distinct).

findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k-1, m-j*j), return(1)))));
a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));

a(n) <= 4. Proof: With Lagrange's four-square theorem, if 4*n is not the sum of 4 positive squares (see A000534), then it is easy to express 3*n as the sum of 3 positive squares. - Yifan Xie and Thomas Scheuerle, Apr 29 2023

A307510 a(n) is the greatest product ijk*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 6, 16, 0, 12, 4, 9, 24, 8, 18, 0, 16, 36, 12, 32, 0, 24, 54, 0, 48, 20, 36, 81, 40, 72, 30, 64, 0, 60, 108, 45, 96, 40, 90, 48, 80, 144, 60, 135, 72, 120, 54, 0, 192, 108, 180, 96, 160, 72, 162, 256, 144, 240, 100

Offset: 0

Rémy Sigrist, Apr 11 2019

The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.

			For n = 34:
- 34 can be expressed in 4 ways as a sum of four squares:
    i^2 + j^2 + k^2 + l^2   i*j*k*l
    ---------------------   -------
    0^2 + 0^2 + 3^2 + 5^2         0
    0^2 + 3^2 + 3^2 + 4^2         0
    1^2 + 1^2 + 4^2 + 4^2        16
    1^2 + 2^2 + 2^2 + 5^2        20
- a(34) = max(0, 16, 20) = 20.

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of the first 20000 terms (where the color is function of the parity of n)
Rémy Sigrist, C program for A307510
Wikipedia, Lagrange's four-square theorem

See A307531 for the additive variant.

Cf. A000534, A002635, A122922.

C
```
See Links section.
```

g:= proc(n, k) option remember; local a;
  if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
  max(0,seq(a*procname(n-a^2, k-1), a=1..floor(sqrt(n))))
end proc:
seq(g(n, 4), n=0..100); # Robert Israel, Apr 15 2019

Array[Max[Times @@ # & /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)

a(n) = 0 iff n belongs to A000534.

a(n) <= (n/4)^2, with equality if and only if n is an even square. - Robert Israel, Apr 15 2019

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

Original entry on oeis.org

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

Offset: 0

Felix Huber, Jul 22 2025

nonn

a(n) is the number of partitions of n into 4 nonzero squares < n/2.

			The a(51) = 1 multiset is [1, 9, 16, 25].
The a(52) = 3 multisets are [1, 1, 25, 25], [4, 16, 16, 16] and [9, 9, 9, 25].

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A000290, A000414, A000534, A025428, A062890, A385736.

# After Alois P. Heinz (A025428)
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:
A385860:=n->b(n,floor(sqrt((n-1)/2)),4):
seq(A385860(n),n=0..87);

a(n) <= A025428(n).

A123069 Odd positive integers that are not the sum of four positive squares.

Original entry on oeis.org

1, 3, 5, 9, 11, 17, 29, 41

Offset: 1

N. J. A. Sloane, Sep 27 2006

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.

Odd terms of A000534.

sm4=Union[Total/@Tuples[Range[10]^2,4]];Select[Range[1,100,2],!MemberQ[sm4,#]&] (* James C. McMahon, Nov 15 2024 *)

A330708 Numbers that are not the sum of 2 nonzero squares and a positive cube.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 12, 15, 17, 20, 22, 23, 24, 31, 36, 39, 43, 50, 55, 57, 63, 65, 70, 71, 78, 87, 94, 103, 111, 113, 115, 119, 120, 134, 139, 141, 148, 160, 167, 169, 185, 204, 211, 254, 263, 267, 279, 283, 286, 302, 311, 312, 331, 335, 342, 349, 379, 391

Offset: 1

XU Pingya, Jun 08 2020

nonn

A022552 is a subsequence.

a(490) = A022552(434) = 5042631. No more terms <= 4 * 10^7.

Robert Israel, Table of n, a(n) for n = 1..490

Cf. A022552, A000534, A004214.

N:= 500: # for terms <= N
G1:= add(x^(i^2), i=1..floor(sqrt(N))):
G2:= add(x^(i^3), i=1..floor(N^(1/3))):
G:= expand(G1^2*G2):
select(t -> coeff(G,x,t)=0, [$0..N]); # Robert Israel, Jun 12 2020

m = 0;
n = 400.;
t = Union@Flatten@Table[x^2 + y^2 + z^3, {x, (n/2)^(1/2)}, {y, x, (n - x^2)^(1/2)}, {z, If[x^2 + y^2 < m, Floor[(m - 1 - x^2 - y^2)^(1/3)] + 1, 1], (n - x^2 - y^2)^(1/3)}];
Complement[Range[m, n], t]

cp's OEIS Frontend

A000414 Numbers that are the sum of 4 nonzero squares.

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A025428 Number of partitions of n into 4 nonzero squares.

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Maple

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A003334 Numbers that are the sum of 11 positive cubes.

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A047701 All positive numbers that are not the sum of 5 nonzero squares.

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

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A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

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A307510 a(n) is the greatest product i*j*k*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

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A307510 a(n) is the greatest product ijk*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.