A003336 -id:A003336 - cp's OEIS frontend

A186650 Total number of n-digit numbers requiring 2 positive biquadrates in their representation as sum of biquadrates.

1, 4, 9, 29, 100, 317, 1007, 3146, 10016, 31712, 100204, 316799, 1002314, 3169309, 10022310, 31693094

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + a(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003336, A186649.

isbiquadrate:=proc(n) type(root(n,4),posint); end:
isA003336:=proc(n) local x,y4; if isbiquadrate(n) then false; else for x from 1 do y4:=n-x^4; if y4A003336(k) then i:=i+1; fi; od: return(i); end: for n from 1 do print(a(n)); od;

a(n) = A186649(n)-A186649(n-1).

a(6) from Martin Renner, Feb 26 2011

a(7)-a(16) from Giovanni Resta, Apr 29 2016

A004831 Numbers that are the sum of at most 2 nonzero 4th powers.

Original entry on oeis.org

0, 1, 2, 16, 17, 32, 81, 82, 97, 162, 256, 257, 272, 337, 512, 625, 626, 641, 706, 881, 1250, 1296, 1297, 1312, 1377, 1552, 1921, 2401, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4096, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6561, 6562, 6577, 6642

Offset: 1

N. J. A. Sloane

nonn

Apart from 0, 1, 2, there are no three consecutive terms up to 10^16. The first two consecutive terms not of the form n^4, n^4+1 are 3502321 = 25^4 + 42^4, 3502322 = 17^4 + 43^4. - Charles R Greathouse IV, Oct 17 2017

T. D. Noe, Table of n, a(n) for n = 1..1000

Subsequences include A003336, A000583 and A002645.

a004831 n = a004831_list !! (n-1)
a004831_list = [x ^ 4 + y ^ 4 | x <- [0..], y <- [0..x]]
-- Reinhard Zumkeller, Jul 15 2013

Reap[For[n = 0, n < 10000, n++, If[MatchQ[ PowersRepresentations[n, 2, 4], {{, },_}], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 30 2017 *)

is(n)=#thue(thueinit(z^4+1),n) \\ Ralf Stephan, Oct 18 2013

list(lim)=my(v=List(),t); for(m=0,sqrtnint(lim\=1,4), for(n=0, min(sqrtnint(lim-m^4,4),m), listput(v,n^4+m^4))); Set(v) \\ Charles R Greathouse IV, Sep 28 2015

Call f(x) the number of terms if this sequence up to x. Then x^(7/16) << f(x) << x^(1/2); in other words, n^2 << a(n) << n^(16/7). The upper bound becomes O(n^2) if A230562 is finite. - Charles R Greathouse IV, Jul 12 2024

A088687 Numbers that can be represented as j^4 + k^4, with 0 < j < k, in exactly one way.

Original entry on oeis.org

17, 82, 97, 257, 272, 337, 626, 641, 706, 881, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8962, 10001, 10016, 10081, 10256, 10625, 10657, 11296

Offset: 1

Cino Hilliard, Nov 22 2003

nonn

			17 = 1^4 + 2^4.
635318657 = 133^4 + 134^4 is absent because it is also 59^4 + 158^4 (see A046881, A230562)

Robert Israel, Table of n, a(n) for n = 1..4500

Cf. A003336, A088728.

N:= 2*10^4: # for terms <= N
V:= Vector(N):
for j from 1 while 2*j^4 < N do
  for k from j+1 do
    r:= j^4 + k^4;
    if r > N then break fi;
    V[r]:= V[r]+1;
od od:
select(t -> V[t] = 1, [$1..N]); $ Robert Israel, Dec 15 2019

lst={};Do[Do[x=a^4;Do[y=b^4;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/4)],a+1,-1}],{a,Floor[n^(1/4)],1,-1}],{n,4*7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)

powers2(m1,m2,p1) = { for(k=m1,m2, a=powers(k,p1); if(a==1,print1(k",")) ); } powers(n,p) = { z1=0; z2=0; c=0; cr = floor(n^(1/p)+1); for(x=1,cr, for(y=x+1,cr, z1=x^p+y^p; if(z1 == n,c++); ); ); return(c) }

Edited by Don Reble, May 03 2006

A018786 Numbers that are the sum of two 4th powers in more than one way.

Original entry on oeis.org

635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752

Offset: 1

David W. Wilson

nonn

Since 4th powers are squares, this is a subsequence of A024508, the analog for squares. Sequence A001235 is the analog for third powers (taxicab numbers). Sequence A255351 lists max {a,b,c,d} where a^4 + b^4 = c^4 + d^4 = a(n), while A255352 lists the whole quadruples (a,b,c,d). - M. F. Hasler, Feb 21 2015

			a(1) = 59^4 + 158^4 = 133^4 + 134^4.
a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - _M. F. Hasler_, Feb 21 2015

R. K. Guy, Unsolved Problems in Number Theory, D1.

Mia Muessig, Table of n, a(n) for n = 1..30000 (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
Mia Muessig, Julia code for finding general taxicab numbers
Eric Weisstein's World of Mathematics, Biquadratic Number.
Eric Weisstein's World of Mathematics, Diophantine Equation.

Subsequence of A003336 (and hence A004831) and A024508 (and hence A001481).

Cf. A001235, A003336, A003824, A309762.

Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* Jean-François Alcover, Jul 26 2011 *)

n=4;L=[];for(b=1,999,for(a=1,b,t=a^n+b^n;for(c=a+1,sqrtn(t\2,n),ispower(t-c^n,n)||next;print1(t",")))) \\ M. F. Hasler, Feb 21 2015

list(lim)=my(v=List()); for(a=134,sqrtnint(lim,4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a,4)+1,min(sqrtnint(lim-a4,4),a), my(t=a4+b^4); for(c=a+1,sqrtnint(lim,4), if(ispower(t-c^4,4), listput(v,t); break)))); Set(v) \\ Charles R Greathouse IV, Jul 12 2024

A weak lower bound: a(n) >> n^2. - Charles R Greathouse IV, Jul 12 2024

A085319 Primes which are the sum of three 5th powers.

Original entry on oeis.org

3, 307, 487, 9043, 16871, 17293, 17863, 23057, 32359, 32801, 33857, 36739, 40787, 43669, 50599, 59051, 59113, 62417, 65537, 76099, 101267, 104149, 107777, 135893, 160073, 161053, 164419, 249107, 249857, 256609, 259733, 266663, 338909, 340649

Offset: 1

Labos Elemer, Jul 01 2003

nonn

Primes in the sumset {A000584 + A000584 + A000584}. There must be an odd number of odd terms in the sum, either 3 odd terms (as with 3 = 1^5 + 1^5 + 1^5 and 487 = 1^5 + 3^5 + 3^5 and 59051 = 1^5 + 1^5 + 9^5) or two even terms and one odd term (as with 307 = 2^5 + 2^5 + 3^5 and 9043 = 3^5 + 4^5 + 6^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime. - Jonathan Vos Post, Sep 24 2006

			a(1) = 3 = 1^5 + 1^5 + 1^5.
a(2) = 307 = 2^5 + 2^5 + 3^5.
a(3) = 487 = 1^5 + 3^5 + 3^5.
a(4) = 9043 = 3^5 + 4^5 + 6^5.
a(5) = 16871 = 2^5 + 2^5 + 7^5.
a(6) = 17293 = 3^5 + 3^5 + 7^5.

Vladimir Joseph Stephan Orlovsky, Table of n, a(n) for n = 1..1000

Cf. A003348, A007490, A085318.

Cf. A000040, A000584, A003336, A003347.

lim = 10^6; nn = Floor[(lim - 2)^(1/5)]; t = {}; Do[p = i^5 + j^5 + k^5; If[p <= lim && PrimeQ[p], AppendTo[t, p]], {i, nn}, {j, i}, {k, j}]; t = Union[t] (* Vladimir Joseph Stephan Orlovsky and T. D. Noe, Jul 15 2011 *)
Select[Prime[Range[2,30000]],Length[PowersRepresentations[#,3,5]]>0&] (* Harvey P. Dale, Nov 26 2014 *)

A123032 was identical. - T. D. Noe, Jul 15 2011

A344187 Numbers that are the sum of two positive fourth powers in exactly one way.

Original entry on oeis.org

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, 10657, 11296, 12401, 13122, 14096, 14642

Offset: 1

David Consiglio, Jr., May 11 2021

nonn

Differs from A003336 at term 11660 because 635318657 = 59^4 + 158^4 = 133^4 + 134^4

			32 is a member of this sequence because 32 = 2^4 + 2^4

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A003336, A338667, A344188.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1,50)]
for pos in cwr(power_terms,2):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v == 1])
for x in range(len(rets)):
    print(rets[x])

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

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)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

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).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

Incorrect program removed by David A. Corneth, Aug 01 2020

A060387 Numbers k such that x^4 + y^4 = k * z^4 is solvable in nonzero integers x,y,z.

Original entry on oeis.org

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, 5906

Offset: 1

Michel ten Voorde, Apr 04 2001

Numbers k such that k = x^4 + y^4 has a solution in positive rationals x, y.

A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
Steven R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Eric Weisstein's World of Mathematics, Biquadratic Number

5906 is the first term not in A003336. Cf. A020897.

Cf. A111152, A209431.

r[n_, z_] := Reduce[0 < x <= y && x^4 + y^4 == n*z^4, {x, y}, Integers]; zm[] = 1; zm[5906] = 17; ok[n] := (tf = False; Do[ If[ r[n, z] =!= False, tf = True; Break[]], {z, 1, zm[n]}]; tf); A060387 = Reap[ Do[ If[ ok[n], Print[n]; Sow[n]], {n, 1, 5906}]][[2, 1]](* Jean-François Alcover, Mar 09 2012 *)

A123294 Sum of 13 positive 5th powers.

Original entry on oeis.org

13, 44, 75, 106, 137, 168, 199, 230, 255, 261, 286, 292, 317, 323, 348, 354, 379, 385, 410, 416, 441, 472, 497, 503, 528, 534, 559, 565, 590, 596, 621, 627, 652, 683, 714, 739, 745, 770, 776, 801, 807, 832, 838, 863, 894, 925, 956, 981, 987, 1012, 1018, 1036

Offset: 1

Jonathan Vos Post, Sep 24 2006

Up to 416 = 13*(2^5) this sequence is identical to x+1 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (13, 137, 199, 317, ...) are A123299. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.

			a(1) = 13 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 44 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(9) = 255 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(11) = 286 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123295-A123297, A008480, A123299.

up = 1500; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 13}]; a (* Giovanni Resta, Jun 12 2016 *)

Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

Two missing terms and more terms from Giovanni Resta, Jun 12 2016

A123295 Sum of 14 positive 5th powers.

Original entry on oeis.org

14, 45, 76, 107, 138, 169, 200, 231, 256, 262, 287, 293, 318, 324, 349, 355, 380, 386, 411, 417, 442, 448, 473, 498, 504, 529, 535, 560, 566, 591, 597, 622, 628, 653, 659, 684, 715, 740, 746, 771, 777, 802, 808, 833, 839, 864, 870, 895, 926, 957, 982, 988

Offset: 1

Jonathan Vos Post, Sep 24 2006

Up to 417 = 13*(2^5) + 1 this sequence is identical to x+2 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (107, 293, 349, 653, ...) are A123300. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.

			a(1) = 14 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 45 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(9) = 256 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(11) = 287 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294-A123297, A008480, A123299, A123300.

up = 1000; q = Range[up^(1/5)]^5; a ={0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a=b, {k, 14}]; a (* Giovanni Resta, Jun 12 2016 *)

Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

5 missing terms and more terms from Giovanni Resta, Jun 12 2016

cp's OEIS Frontend

A186650 Total number of n-digit numbers requiring 2 positive biquadrates in their representation as sum of biquadrates.

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A004831 Numbers that are the sum of at most 2 nonzero 4th powers.

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A088687 Numbers that can be represented as j^4 + k^4, with 0 < j < k, in exactly one way.

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A018786 Numbers that are the sum of two 4th powers in more than one way.

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A085319 Primes which are the sum of three 5th powers.

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A344187 Numbers that are the sum of two positive fourth powers in exactly one way.

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A003386 Numbers that are the sum of 8 nonzero 8th powers.

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