A004812 -id:A004812 - cp's OEIS frontend

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

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

A004823 Numbers that are the sum of 12 positive 11th powers.

Original entry on oeis.org

12, 2059, 4106, 6153, 8200, 10247, 12294, 14341, 16388, 18435, 20482, 22529, 24576, 177158, 179205, 181252, 183299, 185346, 187393, 189440, 191487, 193534, 195581, 197628, 199675, 354304, 356351, 358398, 360445, 362492, 364539, 366586, 368633, 370680, 372727, 374774

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 04 2020: (Start)
208428902 is in the sequence as 208428902 = 1^11 + 2^11 + 3^11 + 3^11 + 3^11 + 4^11 + 4^11 + 4^11 + 5^11 + 5^11 + 5^11 + 5^11.
562491247 is in the sequence as 562491247 = 2^11 + 2^11 + 2^11 + 2^11 + 2^11 + 3^11 + 4^11 + 5^11 + 5^11 + 5^11 + 5^11 + 6^11.
620052034 is in the sequence as 620052034 = 3^11 + 3^11 + 3^11 + 4^11 + 4^11 + 4^11 + 5^11 + 5^11 + 5^11 + 5^11 + 5^11 + 6^11. (End)

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

Cf. A008455 (eleventh powers), A003335 - A004812 (same for 3rd - 10th powers).

Select[Union[Total[#^11]&/@Tuples[Range[3],{12}]],#<+400000&]  (* Harvey P. Dale, Apr 29 2011 *)

A004823_upto(N, n=12, p=11)=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

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

A008454 Tenth powers: a(n) = n^10.

Original entry on oeis.org

0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, 289254654976, 576650390625, 1099511627776, 2015993900449, 3570467226624, 6131066257801, 10240000000000, 16679880978201, 26559922791424

Offset: 0

N. J. A. Sloane

Fifth powers of the squares and the squares of fifth powers. - Wesley Ivan Hurt, Apr 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

a(n) = A123867(n) + 1.
Cf. A000290 (n^2), A000584 (n^5), A013668.
Cf. A004802 - A004812 (sums of 2, ..., 12 nonzero tenth powers).

[n^10: n in [0..20]]; // Vincenzo Librandi, Jun 20 2011

A008454:=n->n^10; seq(A008454(n), n=0..20); # Wesley Ivan Hurt, Jan 22 2014

Table[n^10,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)

A008454(n)=n^10 \\ M. F. Hasler, Jul 03 2025

A008454 = lambda n: n**10 # M. F. Hasler, Jul 03 2025

Multiplicative with a(p^e) = p^(10e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^10 for primes p. - Jaroslav Krizek, Nov 01 2009
From Robert Israel, Mar 31 2016: (Start)
G.f.: x*(x + 1)*(x^8 + 1012*x^7 + 46828*x^6 + 408364*x^5 + 901990*x^4 + 408364*x^3 + 46828*x^2 + 1012*x + 1)/(1 - x)^11.
E.g.f.: x*exp(x)*(x^9 + 45*x^8 + 750*x^7 + 5880*x^6 + 22827*x^5 + 42525*x^4 + 34105*x^3 + 9330*x^2 + 511*x + 1). (End)
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(10) = Pi^10/93555 (A013668).
Sum_{n>=1} (-1)^(n+1)/a(n) = 511*zeta(10)/512 = 73*Pi^10/6842880. (End)

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

A003368 Numbers that are the sum of 12 positive 6th powers.

Original entry on oeis.org

12, 75, 138, 201, 264, 327, 390, 453, 516, 579, 642, 705, 740, 768, 803, 866, 929, 992, 1055, 1118, 1181, 1244, 1307, 1370, 1433, 1468, 1531, 1594, 1657, 1720, 1783, 1846, 1909, 1972, 2035, 2098, 2196, 2259, 2322, 2385, 2448, 2511, 2574, 2637, 2700, 2763, 2924, 2987

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
54710 is in the sequence as 54710 = 2^6 + 3^6 + 3^6 + 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 4^6 + 4^6 + 5^6 + 5^6.
94302 is in the sequence as 94302 = 1^6 + 1^6 + 1^6 + 1^6 + 1^6 + 2^6 + 2^6 + 2^6 + 2^6 + 3^6 + 6^6 + 6^6.
133585 is in the sequence as 133585 = 1^6 + 1^6 + 1^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 7^6. (End)

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

Cf. A001014 (sixth powers).

Cf. A003358 - A003367 (numbers that are the sum of 2, ..., 11 positive 6th powers); A003335, A003346, A003357, A003379, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

Module[{upto=2200,r},r=Ceiling[Surd[upto,6]];Select[Union[Total/@ Tuples[ Range[r]^6,12]],#<=upto&]] (* Harvey P. Dale, Aug 25 2015 *)

(A003368_upto(N, k=12, m=6)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(3000) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

A003379 Numbers that are the sum of 12 positive 7th powers.

Original entry on oeis.org

12, 139, 266, 393, 520, 647, 774, 901, 1028, 1155, 1282, 1409, 1536, 2198, 2325, 2452, 2579, 2706, 2833, 2960, 3087, 3214, 3341, 3468, 3595, 4384, 4511, 4638, 4765, 4892, 5019, 5146, 5273, 5400, 5527, 5654, 6570, 6697, 6824, 6951, 7078, 7205, 7332, 7459, 7586, 7713

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
367634 is in the sequence as 367634 = 3^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 5^7 + 5^7.
456545 is in the sequence as 456545 = 2^7 + 2^7 + 2^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 5^7 + 5^7 + 5^7 + 5^7.
755708 is in the sequence as 755708 = 1^7 + 2^7 + 2^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 5^7 + 5^7 + 6^7 + 6^7. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001015 (seventh powers).

Cf. A003369 - A003378 (numbers that are the sum of 2, ..., 11 positive 7th powers); A003335, A003346, A003357, A003368, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

(A003379_upto(N, k=12, m=7)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(8000) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

Offset corrected by David A. Corneth, Aug 03 2020

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

A003338 Numbers that are the sum of 4 nonzero 4th powers.

Original entry on oeis.org

4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153, 1218

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)
53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4.
81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4.
106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (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. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero 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).

# returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d.
A003338 := proc(n)
    local a,i,j,k,l,res ;
    a := 0 ;
    for i from 1 do
        if i^4 > n then
            break ;
        end if;
        for j from i do
            if i^4+j^4 > n then
                break ;
            end if;
            for k from j do
                if i^4+j^4+k^4> n then
                    break;
                end if;
                res := n-i^4-j^4-k^4 ;
                if issqr(res) then
                    res := sqrt(res) ;
                    if issqr(res) then
                        l := sqrt(res) ;
                        if l >= k then
                            a := a+1 ;
                        end if;
                    end if;
                end if;
            end do:
        end do:
    end do:
    a ;
end proc:
for n from 1 do
    if A003338(n) > 0 then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 17 2023

f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]],seq}, seq=Union[Total/@(Tuples[Range[nn],{4}]^4)]; Select[seq,#<=maxno&]]
f[1000] (* Harvey P. Dale, Feb 27 2011 *)

limit = 1218
from functools import lru_cache
qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
qds = set(qd)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n, )} if n in qds else set()
  return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # Michael S. Branicky, Apr 19 2021

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