A000408 -id:A000408 - cp's OEIS frontend

A003336 Numbers that are the sum of 2 positive 4th powers.

2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8192, 8962, 10001, 10016, 10081, 10256, 10625

Offset: 1

N. J. A. Sloane

Numbers k such that k = x^4 + y^4 has a solution in positive integers x, y.
There are no squares in this sequence. - Altug Alkan, Apr 08 2016
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
16378801 is in the sequence as 16378801 = 43^4 + 60^4.
39126977 is in the sequence as 39126977 = 49^4 + 76^4.
71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)

Sean A. Irvine, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..10000 from David A. Corneth)
A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
S. R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On Generalized Fermat-Wiles Equation [From the Wayback Machine]
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
Eric Weisstein's World of Mathematics, Biquadratic Number.

5906 is the first term in A060387 but not in this sequence. Cf. A020897.
Cf. A088687 (2 distinct 4th powers).
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).
Cf. A000583 (4th powers).

nn=12; Select[Union[Plus@@@(Tuples[Range[nn],{2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)
Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 08 2016 *)

list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1,4), for(y=1, min(sqrtnint(lim-x^4,4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024

T=thueinit('x^4+1,1);
is(n)=#thue(T,n)>0 && !issquare(n) \\ Charles R Greathouse IV, Feb 26 2017

def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  return sorted(p2)
print(aupto(10625)) # Michael S. Branicky, Mar 18 2021

{i: A216284(i) > 0}. - R. J. Mathar, Jun 04 2021

A003327 Numbers that are the sum of 4 positive cubes in 1 or more way.

Original entry on oeis.org

4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 81, 82, 88, 89, 93, 100, 107, 108, 119, 126, 128, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 217, 219, 224, 226, 233, 240, 243, 245, 252, 254

Offset: 1

N. J. A. Sloane

It is conjectured that every number greater than 7373170279850 is in this sequence. [See the paper of the same name. - T. D. Noe, May 25 2017] - Charles R Greathouse IV, Jan 14 2017

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)
3888 is in the sequence as 3888 = 6^3 + 6^3 + 12^3 + 12^3.
7729 is in the sequence as 7729 = 2^3 + 4^3 + 14^3 + 17^3.
7875 is in the sequence as 7875 = 5^3 + 10^3 + 15^3 + 15^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jean-Marc Deshouillers, François Hennecart, Bernard Landreau, 7373170279850, Math. Comp. 69 (2000), pp. 421-439. Appendix by I. Gusti Putu Purnaba.
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A025403, A057905 (complement), A025411 (distinct).
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).

list(lim)=my(v=List(),e=1+lim\1,x='x,t); t=sum(i=1,sqrtnint(e-4,3), x^i^3, O(x^e))^4; for(n=4,lim, if(polcoeff(t,n)>0, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2017

More terms from Eric W. Weisstein

A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

N. J. A. Sloane

nonn

An equivalent definition: numbers of the form x^2 + y^2 + z^2 with x,y,z >= 0.
Bourgain studies "the spatial distribution of the representation of a large integer as a sum of three squares, on the small and critical scale as well as their electrostatic energy. The main results announced give strong evidence to the thesis that the solutions behave randomly. This is in sharp contrast to what happens with sums of two or four or more square." Sums of two nonzero squares are A000404. - Jonathan Vos Post, Apr 03 2012
The multiplicities for a(n) (if 0 <= x <= y <= z) are given as A000164(a(n)), n >= 1. Compare with A005875(a(n)) for integer x, y and z, and order taken into account. - Wolfdieter Lang, Apr 08 2013
a(n)^k is a member of this sequence for any k > 1. - Boris Putievskiy, May 05 2013
The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in a simple cubic lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A004014 for f.c.c. lattice. - Mohammed Yaseen, Nov 06 2022

			a(1) = 0 = 0^2 + 0^2 + 0^2. A005875(0) = 1 = A000164(0).
a(9) = 9 = 0^2 + 0^2 + 3^2 =  1^2 +  2^2 + 2^2. A000164(9) = 2. A000164(9) = 30 = 2*3 + 8*3 (counting signs and order). - _Wolfdieter Lang_, Apr 08 2013

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 37.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section C20.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jean Bourgain, Peter Sarnak and Zeév Rudnick, Local statistics of lattice points on the sphere, arXiv:1204.0134 [math.NT], 2012-2015.
J. W. Cogdell, On sums of three squares, Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 33-44.
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70. See Theorem I.
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91.
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Union of A000290, A000404 and A000408 (common elements).
Union of A000290, A000415 and A000419 (disjunct sets).
Complement of A004215.
Cf. A005875 (number of representations if x, y and z are integers).
Cf. A047449, A034043, A034044, A034045, A034046, A034047, A065883, A072400, A072401, A071374, A002828, A001481, A125084, A000164.

isA000378 := proc(n) # return true or false depending on n being in the list
    local x,y ;
    for x from 0 do
        if 3*x^2 > n then
            return false;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            else
                if issqr(n-x^2-y^2) then
                    return true;
                end if;
            end if;
        end do:
    end do:
end proc:
A000378 := proc(n) # generate A000378(n)
    option remember;
    local a;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA000378(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A000378(n),n=1..100) ; # R. J. Mathar, Sep 09 2015

okQ[n_] := If[EvenQ[k = IntegerExponent[n, 2]], m = n/2^k; Mod[m, 8] != 7, True]; Select[Range[0, 100], okQ] (* Jean-François Alcover, Feb 08 2016, adapted from PARI *)

isA000378(n)=my(k=valuation(n, 2)); if(k%2==0, n>>=k; n%8!=7, 1)

list(lim)=my(v=List(),k,t); for(x=0,sqrtint(lim\=1), for(y=0, min(sqrtint(lim-x^2),x), k=x^2+y^2; for(z=0,min(sqrtint(lim-k), y), listput(v,k+z^2)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

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
print(list(filter(ok, range(84)))) # Michael S. Branicky, Jul 15 2021

from itertools import count, islice
def A000378_gen(): # generator of terms
    return filter(lambda n:n>>2*(bin(n)[:1:-1].index('1')//2) & 7 < 7, count(1))
A000378_list = list(islice(A000378_gen(),30)) # Chai Wah Wu, Jun 27 2022

def A000378(n):
    def f(x): return n-1+sum(((x>>(i<<1))-7>>3)+1 for i in range(x.bit_length()>>1))
    m, k = n-1, f(n-1)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

Legendre: a nonnegative integer is a sum of three squares iff it is not of the form 4^k m with m == 7 (mod 8).
n^(2k+1) is in the sequence iff n is in the sequence. - Ray Chandler, Feb 03 2009
Complement of A004215; complement of A000302(i)*A004771(j), i,j>=0. - Boris Putievskiy, May 05 2013
a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014

More terms from Ray Chandler, Sep 05 2004

A000414 Numbers that are the sum of 4 nonzero squares.

Original entry on oeis.org

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

A003328 Numbers that are the sum of 5 positive cubes.

Original entry on oeis.org

5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 64, 68, 71, 75, 78, 82, 83, 89, 90, 94, 96, 97, 101, 108, 109, 115, 116, 120, 127, 129, 131, 134, 135, 136, 138, 143, 145, 146, 150, 152, 153, 155, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 190, 192, 194, 195, 199

Offset: 1

N. J. A. Sloane

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

It seems only a finite number N of positive integers are not in this sequence, and thus a(n) = n - N for all sufficiently large n. Is it true that 2243453, last term of A048927, is sufficiently large in that sense? - M. F. Hasler, Jan 04 2023

			From _David A. Corneth_, Aug 01 2020: (Start)
3084 is in the sequence as 3084 = 5^3 + 5^3 + 5^3 +  8^3 + 13^3.
4385 is in the sequence as 4385 = 4^3 + 4^3 + 9^3 + 11^3 + 13^3.
5426 is in the sequence as 5426 = 8^3 + 9^3 + 9^3 + 12^3 + 12^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003328, A048926, A048297.
Cf. A057906 (Complement)
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
  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).

select( {is_A003328(n,k=5,m=3,L=sqrtnint(abs(n-k+1),m))=if( n>k*L^m || nM. F. Hasler, Aug 02 2020
A003328_upto(N,k=5,m=3)=[i|i<-[1..#N=sum(n=1,sqrtnint(N,m),'x^n^m,O('x^N))^k], polcoef(N,i)] \\ M. F. Hasler, Aug 02 2020

from collections import Counter
from itertools import combinations_with_replacement as combs_w_rep
def aupto(lim):
  s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
  s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_w_rep(s, 5)))
  s2counts = Counter(s2)
  return sorted(k for k in s2counts)
print(aupto(200)) # Michael S. Branicky, May 12 2021

A002479 Numbers of the form x^2 + 2*y^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 19, 22, 24, 25, 27, 32, 33, 34, 36, 38, 41, 43, 44, 48, 49, 50, 51, 54, 57, 59, 64, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 98, 99, 100, 102, 107, 108, 113, 114, 118, 121, 123, 128, 129, 131

Offset: 1

N. J. A. Sloane

A positive number k belongs to this sequence if and only if every prime p == 5, 7 (mod 8) dividing k occurs to an even power. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002
Norms of numbers in Z[sqrt(-2)]. - Alonso del Arte, Sep 23 2014
Euler (E256) shows that these numbers are closed under multiplication, according to the Euler Archive. - Charles R Greathouse IV, Jun 16 2016
In addition to the previous comment: The proof was already given 1100 years before Euler by Brahmagupta's identity (a^2 + m*b^2)*(c^2 + m*d^2) = (a*c - m*b*d)^2 + m*(a*d + b*c)^2. - Klaus Purath, Oct 07 2023

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 421.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..3148 (first 1000 terms from T. D. Noe)
L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
L. Euler, (E256) Specimen de usu observationum in mathesi pura, Novi Commentarii academiae scientiarum Petropolitanae 6 (1761), pp. 185-230.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Complement of A097700. Subsequence of A000408. For primes see A033203.

Cf. A035251, A027748, A124010, A003628.

a002479 n = a002479_list !! (n-1)
a002479_list = 0 : filter f [1..] where
   f x = all (even . snd) $ filter ((`elem` [5,7]) . (`mod` 8) . fst) $
                            zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Feb 20 2014

[n: n in [0..131] | NormEquation(2, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

lis:={}; M:=50; M2:=M^2;
for x from 0 to M do for y from 0 to M do
if x^2+2*y^2 <= M2 then lis:={op(lis),x^2+2*y^2}; fi; od: od:
sort(convert(lis,list)); # N. J. A. Sloane, Apr 30 2015

q = 16; imax = q^2; Select[Union[Flatten[Table[x^2 + 2y^2, {y, 0, q/Sqrt[2]}, {x, 0, q}]]], # <= imax &] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
Union[#[[1]]+2#[[2]]&/@Tuples[Range[0,10]^2,2]] (* Harvey P. Dale, Nov 24 2014 *)

is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,1]%8>4 && f[i,2]%2, return(0)));1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=List()); for(a=0,sqrtint(lim\=1), for(b=0,sqrtint((lim-a^2)\2), listput(v,a^2+2*b^2))); Set(v) \\ Charles R Greathouse IV, Jun 16 2016

from itertools import count, islice
from sympy import factorint
def A002479_gen(): # generator of terms
    return filter(lambda n:all(p & 7 < 5 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A002479_list = list(islice(A002479_gen(),30)) # Chai Wah Wu, Jun 27 2022

A003337 Numbers n which are the sum of 3 nonzero 4th powers.

Original entry on oeis.org

3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177

Offset: 1

N. J. A. Sloane

Numbers which are in this sequence but not in A047714 must also be the sum of 2 biquadrates, or equal to a fourth power. Among the first 1000 terms of this sequence, this is the case for 4802 = 2*7^4, 57122 = 2*13^4 and 76832 = 2*14^4. - M. F. Hasler, Dec 31 2012
The union of A047714, A336536, and fourth powers of A003294. - Robert Israel, Jul 24 2020
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
194818 is in the sequence as 194818 = 3^4 + 4^4 + 21^4.
480113 is in the sequence as 480113 = 7^4 + 12^4 + 26^4.
693842 is in the sequence as 693842 = 13^4 + 15^4 + 28^4. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A003294, A047714, A336536, A309762, A344188.
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).

def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
  return sorted(p3)
print(aupto(2400)) # Michael S. Branicky, Mar 18 2021

A003329 Numbers that are the sum of 6 positive cubes.

Original entry on oeis.org

6, 13, 20, 27, 32, 34, 39, 41, 46, 48, 53, 58, 60, 65, 67, 69, 72, 76, 79, 83, 84, 86, 90, 91, 95, 97, 98, 102, 104, 105, 109, 110, 116, 117, 121, 123, 124, 128, 130, 132, 135, 136, 137, 139, 142, 143, 144, 146, 147, 151, 153, 154, 156, 158, 160, 161, 162, 163, 165, 170

Offset: 1

N. J. A. Sloane

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)
1647 is in the sequence as 1647 = 3^3 + 3^3 + 5^3 + 5^3 +  7^3 + 10^3.
3319 is in the sequence as 3319 = 5^3 + 5^3 + 5^3 + 6^3 + 10^3 + 12^3.
4038 is in the sequence as 4038 = 3^3 + 3^3 + 6^3 + 8^3 +  8^3 + 14^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A048929, A048930, A048931.
Cf. A057907 (Complement)
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
  A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
  A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
  A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
  A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
  A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
  A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

max = 200; cmax = Ceiling[(max - 5)^(1/3)]; cc = Array[c, 6]; iter = Sequence @@ Transpose[ {cc, Join[{1}, Most[cc]], Table[cmax, {6}]}]; Union[ Reap[ Do[ a = Total[cc^3]; If[a <= max, Sow[a]], Evaluate[iter]]][[2, 1]]] (* Jean-François Alcover, Oct 23 2012 *)

(A003329_upto(N,k=6,m=3)=[i|i<-[1..#N=sum(n=1,sqrtnint(N,m), 'x^n^m, O('x^N))^k], polcoef(N,i)])(200) \\ M. F. Hasler, Aug 02 2020

from collections import Counter
from itertools import combinations_with_replacement as multi_combs
def aupto(lim):
  c = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
  s = filter(lambda x: x<=lim, (sum(mc) for mc in multi_combs(c, 6)))
  counts = Counter(s)
  return sorted(k for k in counts)
print(aupto(170)) # Michael S. Branicky, Jun 13 2021

More terms from Eric W. Weisstein

A003346 Numbers that are the sum of 12 positive 4th powers.

Original entry on oeis.org

12, 27, 42, 57, 72, 87, 92, 102, 107, 117, 122, 132, 137, 147, 152, 162, 167, 172, 177, 182, 187, 192, 197, 202, 212, 217, 227, 232, 242, 247, 252, 257, 262, 267, 277, 282, 292, 297, 307, 312, 322, 327, 332, 342, 347, 357, 362, 372, 377, 387, 392, 402, 407, 412, 417

Offset: 1

N. J. A. Sloane

a(88) = 636 = 5^4 + 11 and a(91) = 651 = 5^4 + 2^4 + 10 are the first two terms not congruent to 2 or 7 (mod 10). - M. F. Hasler, Aug 03 2020

			From _David A. Corneth_, Aug 03 2020: (Start)
3740 is in the sequence as 3740 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 3^4 + 5^4 + 5^4 + 7^4.
4690 is in the sequence as 4690 = 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 4^4 + 4^4 + 4^4 + 5^4 + 5^4 + 6^4 + 6^4.
7193 is in the sequence as 7193 = 2^4 + 4^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 6^4. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000583 (4th powers).
Other numbers that are the sum of k positive m-th powers:
  A000404 (k=2, m=2), A000408 (3, 2), A000414 (4, 2), A047700 (k=5, m=2),
  A003325 (k=2, m=3), A003072 (k=3, m=3), A003327 .. A003335 (k=4..12, m=3),
  A003336 .. A003346 (k=2..12, m=4), A003347 .. A003357 (k=2..12, m=5),
  A003358 .. A003368 (k=2..12, m=6), A003369 .. A003379 (k=2..12, m=7),
  A003380 .. A003390 (k=2..12, m=8), A003391 .. A004801 (k=2..12, m=9),
  A004802 .. A004812 (k=2..12, m=10), A004813 .. A004823 (k=2..12, m=11).

(A003346_upto(N, k=12, m=4)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(500) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(limit):
    qd = takewhile(lambda x: x <= limit, (k**4 for k in count(1)))
    ss = set(sum(c) for c in mc(qd, 12))
    return sorted(s for s in ss if s <= limit)
print(aupto(417)) # Michael S. Branicky, Dec 27 2021

A003358 Numbers that are the sum of 2 nonzero 6th powers.

Original entry on oeis.org

2, 65, 128, 730, 793, 1458, 4097, 4160, 4825, 8192, 15626, 15689, 16354, 19721, 31250, 46657, 46720, 47385, 50752, 62281, 93312, 117650, 117713, 118378, 121745, 133274, 164305, 235298, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 524288

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

			From _David A. Corneth_, Aug 01 2020: (Start)
10069120217 is in the sequence as 10069120217 = 29^6 + 46^6.
139314070233 is in the sequence as 139314070233 = 3^6 + 72^6.
404680615040 is in the sequence as 404680615040 = 22^6 + 86^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.

Cf. A088677 (2 distinct 6th). Supersequence of A106318.
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).

With[{k = 6}, Union@ Map[(#[[1]]^k + #[[2]]^k) &, Tuples[Range[8], {2}]]] (* Michael De Vlieger, Sep 09 2022, after Harvey P. Dale at A004999  *)

Removed incorrect program. David A. Corneth, Aug 01 2020

cp's OEIS Frontend

A003336 Numbers that are the sum of 2 positive 4th powers.

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A003327 Numbers that are the sum of 4 positive cubes in 1 or more way.

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A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2.

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A000414 Numbers that are the sum of 4 nonzero squares.

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A003328 Numbers that are the sum of 5 positive cubes.

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A002479 Numbers of the form x^2 + 2*y^2.

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