A000404 -id:A000404 - cp's OEIS frontend

A003329 Numbers that are the sum of 6 positive cubes.

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

A003380 Numbers that are the sum of 2 nonzero 8th powers.

Original entry on oeis.org

2, 257, 512, 6562, 6817, 13122, 65537, 65792, 72097, 131072, 390626, 390881, 397186, 456161, 781250, 1679617, 1679872, 1686177, 1745152, 2070241, 3359232, 5764802, 5765057, 5771362, 5830337, 6155426, 7444417, 11529602, 16777217, 16777472, 16783777, 16842752

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 01 2020: (Start)
274893519322337 is in the sequence as 274893519322337 = 58^8 + 59^8.
357707312890625 is in the sequence as 357707312890625 = 50^8 + 65^8.
2590188068194497 is in the sequence as 2590188068194497 = 57^8 + 84^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5833 terms from R. J. Mathar, replacing an earlier b-file that was missing terms)

Subsequence of A004875.
Cf. A155468 (2 distinct 8th).
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).

A003380 := proc(nmax::integer)
    local a, x,x8,y,y8 ;
    a := {} ;
    for x from 1 do
        x8 := x^8 ;
        if 2*x8 > nmax then
            break;
        end if;
        for y from x do
            y8 := y^8 ;
            if x8+y8 > nmax then
                break;
            end if;
            if x8+y8 <= nmax then
                a := a  union {x8+y8} ;
            end if;
        end do:
    end do:
    sort(convert(a,list)) ;
end proc:
nmax := 20000000000000000 ;
L:= A003380(nmax) ;
LISTTOBFILE(L,"b003380.txt",1) ; # R. J. Mathar, Aug 01 2020

Total/@Tuples[Range[8]^8,2]//Union (* Harvey P. Dale, Apr 04 2017 *)

list(lim)=my(v=List(), x8); for(x=1, sqrtnint(lim\=1, 8), x8=x^8; for(y=1, min(sqrtnint(lim-x8, 8), x), listput(v, x8+y^8))); Set(v) \\ Charles R Greathouse IV, Aug 22 2017

A003335 Numbers that are the sum of 12 positive cubes.

Original entry on oeis.org

12, 19, 26, 33, 38, 40, 45, 47, 52, 54, 59, 61, 64, 66, 68, 71, 73, 75, 78, 80, 82, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111, 113, 115, 116, 117, 118, 120, 122, 123, 124, 125, 127, 129, 130, 131, 132, 134, 136, 137, 138, 139, 141, 142

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)
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

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. A000578 (cubes).

(A003335_upto(N, k=12, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ Use 2nd & 3rd optional arg to get other sequences of this family. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

Original entry on oeis.org

1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421

Offset: 1

N. J. A. Sloane

Also gives solutions z to x^2+y^2=z^4 with gcd(x,y,z)=1 and x,y,z positive. - John Sillcox (johnsillcox(AT)hotmail.com), Feb 20 2004
A065338(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
Product_{k=1..A001221(a(n))} A079260(A027748(a(n),k)) = 1. - Reinhard Zumkeller, Jan 07 2013
A062327(a(n)) = A000005(a(n))^2. (These are the only numbers that satisfy this equation.) - Benedikt Otten, May 22 2013
Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - Michael Somos, Jul 26 2013
Numbers k such that there is a "knight's move" of Euclidean distance sqrt(k) which allows the whole of the 2D lattice to be reached. For example, a knight which travels 4 units in any direction and then 1 unit at right angles to the first direction moves a distance sqrt(17) for each move. This knight can reach every square of an infinite chessboard.
Also 1/7 of the area of the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 nodes of the square lattice - at its vertices. - Alexander M. Domashenko, Feb 21 2024
Sequence closed under multiplication. Odd values of A031396 and their powers. These are the only numbers m that satisfy the Pell equation (k*x)^2 - D*(m*y)^2 = -1. - Klaus Purath, May 12 2025

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

T. D. Noe, Table of n, a(n) for n = 1..10000
David Broadhurst and Wadim Zudilin, A magnetic double integral, arXiv:1708.02381 [math.NT], 2017. See p. 7
Code Golf Stack Exchange, pseudonymous Stack Exchange user 'trichoplax', N-movers: How much of the infinite board can I reach?

Subsequence of A000404; A002144 is a subsequence. Essentially same as A008846.

Cf. A004614.

a004613 n = a004613_list !! (n-1)
a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
-- Reinhard Zumkeller, Jan 07 2013

[n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // Vincenzo Librandi, Aug 21 2012

isA004613 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if modp(p,4) <> 1 then
            return false;
        end if;
    end do:
    true;
end proc:
for n from 1 to 200 do
    if isA004613(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 17 2014
# second Maple program:
q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
select(q, [$1..500])[];  # Alois P. Heinz, Jan 13 2024

ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-François Alcover, May 05 2011 *)
Select[Range[500],Union[Mod[#,4]&/@(FactorInteger[#][[All,1]])]=={1}&] (* Harvey P. Dale, Mar 08 2017 *)

for(n=1,1000,if(sumdiv(n,d,isprime(d)*if((d-1)%4,1,0))==0,print1(n,",")))

is(n)=n%4==1 && factorback(factor(n)[,1]%4)==1 \\ Charles R Greathouse IV, Sep 19 2016

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.

A003333 Numbers that are the sum of 10 positive cubes.

Original entry on oeis.org

10, 17, 24, 31, 36, 38, 43, 45, 50, 52, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 108, 109, 111, 113, 114, 115, 116, 118, 120, 121, 122, 123, 125, 127, 128, 129, 130, 132, 134, 135, 136, 137, 139, 140, 141, 142

Offset: 1

N. J. A. Sloane

374 is the largest of only 99 positive integers not in this sequence. - M. F. Hasler, Aug 13 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1646 is in the sequence as 1646 = 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 7^3 + 7^3 +  8^3.
2790 is in the sequence as 2790 = 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 6^3 + 6^3 + 7^3 + 8^3 + 10^3.
3450 is in the sequence as 3450 = 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 7^3 + 8^3 + 8^3 + 9^3 +  9^3. (End)

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

Other sequences of numbers that are the sum of x nonzero y-th powers:
  A000404 (x=2, y=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).

(A003333_upto(N)=select( {is_A003333(n,k=10,m=3,L=sqrtnint(abs(n-k+1),m))=if( n>k*L^m || nM. F. Hasler, Aug 02 2020
A3333=A003333_upto(320); A003333(n)=if(n>275, n+99, n>222, n+98, A3333[n]) \\ M. F. Hasler, Aug 13 2020

a(n) = n + 99 for all n > 275. - M. F. Hasler, Aug 13 2020

A003369 Numbers that are the sum of 2 positive 7th powers.

Original entry on oeis.org

2, 129, 256, 2188, 2315, 4374, 16385, 16512, 18571, 32768, 78126, 78253, 80312, 94509, 156250, 279937, 280064, 282123, 296320, 358061, 559872, 823544, 823671, 825730, 839927, 901668, 1103479, 1647086, 2097153, 2097280, 2099339, 2113536

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
3909794986386 is in the sequence as 3909794986386 = 57^7 + 57^7.
6061605477062 is in the sequence as 6061605477062 = 19^7 + 67^7.
26019535290982 is in the sequence as 26019535290982 = 61^7 + 81^7. (End)

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

Cf. A000404 (2 squares), A003325 (2 cubes), A003336 (2 4th), A003347 (2 5th), A003358 (2 6th), A088719 (2 distinct 7th), A003380 (2 8th).

Cf. A001015 (seventh powers).

N:= 10^7: # to get all terms <= N
S:= select(`<=`, {seq(seq(a^7+b^7, a=1..b), b=1..floor(N^(1/7)))}, N):
sort(convert(S, list)); # Robert Israel, Sep 03 2017

lst={}; Do[If[(a^7+b^7)==n, AppendTo[lst, n]], {n, 200000}, {a, (n/2)^(1/7)}, {b, a, (n-a^7)^(1/7)}]; lst (* XU Pingya, Sep 03 2017 *)
Module[{upto=10},Select[Union[Total/@Tuples[Range[upto]^7,2]],#<= (upto^7)&]] (* Harvey P. Dale, Feb 04 2019 *)

A003390 Sum of 12 nonzero 8th powers.

Original entry on oeis.org

12, 267, 522, 777, 1032, 1287, 1542, 1797, 2052, 2307, 2562, 2817, 3072, 6572, 6827, 7082, 7337, 7592, 7847, 8102, 8357, 8612, 8867, 9122, 9377, 13132, 13387, 13642, 13897, 14152, 14407, 14662, 14917, 15172, 15427, 15682, 19692, 19947, 20202, 20457

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)
1890948 is in the sequence as 1890948 = 2^8 + 2^8 + 2^8 + 4^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 5^8 + 5^8 + 5^8.
2338951 is in the sequence as 2338951 = 1^8 + 1^8 + 1^8 + 1^8 + 1^8 + 3^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 6^8.
3841896 is in the sequence as 3841896 = 1^8 + 1^8 + 1^8 + 2^8 + 3^8 + 3^8 + 3^8 + 3^8 + 4^8 + 5^8 + 6^8 + 6^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (replacing a b-file that missed terms)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2), A003072 (3, 3), A003325 (2, 3), 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), A003386 (8, 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).

A003390_upto(N=1e5, n=12, p=8)={my(P=[x^p|x<-[1..sqrtnint(N-n+1, p)]], S=P); while(n--, S=Set(concat([[x+y|y<-S, x+y<=N]|x<-P]))); S} \\ M. F. Hasler, Jul 03 2025

Removed incorrect program, offset corrected by David A. Corneth, Aug 01 2020

A004802 Numbers that are the sum of 2 nonzero 10th powers.

Original entry on oeis.org

2, 1025, 2048, 59050, 60073, 118098, 1048577, 1049600, 1107625, 2097152, 9765626, 9766649, 9824674, 10814201, 19531250, 60466177, 60467200, 60525225, 61514752, 70231801, 120932352, 282475250, 282476273, 282534298, 283523825, 292240874

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)
1103972715709403850 is in the sequence as 1103972715709403850 = 51^10 + 63^10.
2059617246125773226 is in the sequence as 2059617246125773226 = 61^10 + 65^10.
27850192968371852849 is in the sequence as 27850192968371852849 = 25^10 + 88^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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).

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

cp's OEIS Frontend

A003329 Numbers that are the sum of 6 positive cubes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A003380 Numbers that are the sum of 2 nonzero 8th powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A003335 Numbers that are the sum of 12 positive cubes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

A003333 Numbers that are the sum of 10 positive cubes.

Views

Author

Keywords

Comments

Examples

Links