A004831 -id:A004831 - cp's OEIS frontend

A000583 Fourth powers: a(n) = n^4.

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921

Offset: 0

N. J. A. Sloane

Figurate numbers based on 4-dimensional regular convex polytope called the 4-measure polytope, 4-hypercube or tesseract with Schlaefli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
Totally multiplicative sequence with a(p) = p^4 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A058649. The inverse binomial transforms yields the (finite) 0, 1, 14, 36, 24, the 4th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Generate Pythagorean triangles with parameters a and b to get sides of lengths x = b^2-a^2, y = 2*a*b, and z = a^2 + b^2. In particular use a=n-1 and b=n for a triangle with sides (x1,y1,z1) and a=n and b=n+1 for another triangle with sides (x2,y2,z2). Then x1*x2 + y1*y2 + z1*z2 = 8*a(n). - J. M. Bergot, Jul 22 2013
For n > 0, a(n) is the largest integer k such that k^4 + n is a multiple of k + n. Also, for n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n^2. - Derek Orr, Sep 04 2014
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
a(n+2)/2 is the area of a trapezoid with vertices at (T(n), T(n+1)), (T(n+1), T(n)), (T(n+1), T(n+2)), and (T(n+2), T(n+1)) with T(n)=A000292(n) for n >= 0. - J. M. Bergot, Feb 16 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
Dov Juzuk, Curiosa 56: An interesting observation, Scripta Mathematica 6 (1939), 218.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 47.

T. D. Noe, Table of n, a(n) for n = 0..1000
Henry Bottomley, Illustration of initial terms
Henry Bottomley, Some Smarandache-type multiplicative sequences
Ralph Greenberg, Math for Poets.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
Eric Weisstein's World of Mathematics, Biquadratic Number.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index entries for sequences related to Benford's law

Cf. A000538, A005917 (first differences), A000332, A014820, A092181, A092182, A092183.
Cf. A004831, A002646.
Cf. A002593, A260810. - Bruno Berselli, Jul 31 2015
Cf. A002415, A000290, A006008, A132366, A039623, A139584, A071270, A047928, A187756.
Cf. A062392, A231303, A267315.

a000583 = (^ 4)
a000583_list = scanl (+) 0 a005917_list
-- Reinhard Zumkeller, Nov 13 2014, Nov 11 2012

[n^4 : n in [0..50]]; // Wesley Ivan Hurt, Sep 05 2014

A000583 := n->n^4: seq(A000583(n), n=0..50);
A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); # Zerinvary Lajos, May 25 2008

Range[0,100]^4 (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)

makelist(n^4,n,0,30); /* Martin Ettl, Nov 12 2012 */

A000583(n) = n^4 \\ Michael B. Porter, Nov 09 2009

def a(n): return n**4
print([a(n) for n in range(34)]) # Michael S. Branicky, Nov 10 2022

a(n) = A123865(n)+1 = A002523(n)-1.
Multiplicative with a(p^e) = p^(4e). - David W. Wilson, Aug 01 2001
G.f.: x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. More generally, g.f. for n^m is Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292).
Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: (x + 7*x^2 + 6*x^3 + x^4)*e^x. More generally, the general form for the e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial of order n. - Franklin T. Adams-Watters, Sep 11 2005
Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - Jaume Oliver Lafont, Sep 20 2009
a(n) = C(n+3,4) + 11*C(n+2,4) + 11*C(n+1,4) + C(n,4). [Worpitzky's identity for powers of 4. See, e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - Bruno Berselli, May 07 2010
a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - Charlie Marion, Jun 13 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - Ant King, Sep 23 2013
From Amiram Eldar, Jan 20 2021: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).
Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)

A047217 Numbers that are congruent to {0, 1, 2} mod 5.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 30, 31, 32, 35, 36, 37, 40, 41, 42, 45, 46, 47, 50, 51, 52, 55, 56, 57, 60, 61, 62, 65, 66, 67, 70, 71, 72, 75, 76, 77, 80, 81, 82, 85, 86, 87, 90, 91, 92, 95, 96, 97, 100, 101, 102, 105, 106, 107, 110, 111

Offset: 1

N. J. A. Sloane

Also, the only numbers that are eligible to be the sum of two 4th powers (A004831). - Cino Hilliard, Nov 23 2003
Nonnegative m such that floor(2*m/5) = 2*floor(m/5). - Bruno Berselli, Dec 09 2015
The sequence lists the indices of the multiples of 5 in A007531. - Bruno Berselli, Jan 05 2018

Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A007531, A030341, A004831 (two 4th powers).

Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.

I:=[0, 1, 2, 5]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 25 2012

&cat [[5*n,5*n+1,5*n+2]: n in [0..30]]; // Bruno Berselli, Dec 09 2015

seq(op([5*i,5*i+1,5*i+2]),i=0..100); # Robert Israel, Sep 02 2014

Select[Range[0,120], MemberQ[{0,1,2}, Mod[#,5]]&] (* Harvey P. Dale, Jan 20 2012 *)

a(n)=n--\3*5+n%3 \\ Charles R Greathouse IV, Oct 22 2011

concat(0, Vec(x^2*(1+x+3*x^2)/(1-x)^2/(1+x+x^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015

is(n) = n%5 < 3 \\ Felix Fröhlich, Jan 05 2018

a(n+1) = Sum_{k>=0} A030341(n,k)*b(k) with b(0)=1 and b(k)=5*3^(k-1) for k>0. - Philippe Deléham, Oct 22 2011
G.f.: x^2*(1+x+3*x^2)/(1-x)^2/(1+x+x^2). - Colin Barker, Feb 17 2012
a(n) = 5 + a(n-3) for n>3. - Robert Israel, Sep 02 2014
a(n) = floor((5/4)*floor(4*(n-1)/3)). - Bruno Berselli, May 03 2016
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (15*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3*k) = 5*k-3, a(3*k-1) = 5*k-4, a(3*k-2) = 5*k-5. (End)
a(n) = n - 1 + 2*floor((n-1)/3). - Bruno Berselli, Feb 06 2017
Sum_{n>=2} (-1)^n/a(n) = sqrt(1-2/sqrt(5))*Pi/5 + 3*log(2)/5. - Amiram Eldar, Dec 10 2021

A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

Original entry on oeis.org

2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177, 4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561, 28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161, 66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, 121937

Offset: 1

N. J. A. Sloane

The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144 + 1 = (145310^65536)^4 + 1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011

Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017

			a(1) =   2 = 1^4 + 1^4.
a(2) =  17 = 1^4 + 2^4.
a(3) =  97 = 2^4 + 3^4.
a(4) = 257 = 1^4 + 4^4.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 22-8 Mu Alpha Theta 1984.
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).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Subsequence of A002313 and of A028916.
Intersection of A004831 and A000040.
Cf. A002646, A000583, A003336, A000290, A256852.

a002645 n = a002645_list !! (n-1)
a002645_list = 2 : (map a000040 $ filter ((> 1) . a256852) [1..])
-- Reinhard Zumkeller, Apr 11 2015

nn = 100000; Sort[Reap[Do[n = a^4 + b^4; If[n <= nn && PrimeQ[n], Sow[n]], {a, nn^(1/4)}, {b, a}]][[2, 1]]]
With[{nn=20},Select[Union[Flatten[Table[x^4+y^4,{x,nn},{y,nn}]]],PrimeQ[ #] && #<=nn^4+1&]] (* Harvey P. Dale, Aug 10 2021 *)

upto(lim)=my(v=List(2),t);forstep(x=1,lim^.25,2,forstep(y=2,(lim-x^4)^.25,2,if(isprime(t=x^4+y^4),listput(v,t))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 05 2011

list(lim)=my(v=List([2]),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; forstep(y=1+x%2,min(sqrtnint(lim-x4,4), x-1),2, if(isprime(t=x4+y^4), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017

A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006

A256852(A049084(a(n))) > 1 for n > 1. - Reinhard Zumkeller, Apr 11 2015

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002

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

A351306 Least positive integer m such that m^4n = u^4 + v^4 - (x^4 + y^4) for some nonnegative integers u,v,x,y with x^4 + y^4 <= m^4n^2.

Original entry on oeis.org

1, 1, 1, 10, 2, 2, 2, 4, 6, 4, 2, 2, 4, 8, 1, 1, 1, 1, 2, 2, 2, 2, 10, 2, 2, 2, 2, 10, 10, 2, 1, 1, 1, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 10, 2, 2, 1, 1, 5, 1, 1, 4, 10, 10, 2, 2, 6, 10, 4, 4, 2, 4, 1, 3, 1, 1, 1, 10, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 4, 10, 2, 2, 4, 6, 6, 1, 1, 1, 1, 5, 2, 2

Offset: 0

Zhi-Wei Sun, Feb 06 2022

nonn

Conjecture: Each n >= 0 can be written as u^4 + v^4 - (x^4 + y^4), where u,v,x,y are rational numbers with x^4 + y^4 <= n^2. In other words, a(n) exists for any nonnegative integer n.

A known result of R. Norrie states that any rational number can be written as u^4 + v^4 - (x^4 + y^4) with u,v,x,y rational numbers.

			a(3) = 10 with 10^4*3 = 8^4 + 13^4 - (4^4 + 7^4) and 4^4 + 7^4 <= 10^4*3^2.
a(242) = 15 with 15^4*242 = 73^4 + 153^4 - (36^4 + 154^4) and 36^4 + 154^4 <= 15^4*242^2.
a(248) = 28 with 28^4*248 = 95^4 + 270^4 - (52^4 + 269^4) and 52^4 + 269^4 <= 28^4*248^2.
a(313) = 30 with 30^4*313 = 37^4 + 128^4 - (7^4 + 64^4) and 7^4 + 64^4 <= 30^4*313^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..800
Tito Piezas III, A Collection of Algebraic Identities, 001b: Assorted identities, Part 2 (Waring-like problems).
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000583, A004831, A351312.

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
tab={};Do[m=1;Label[bb];k=m^4;Do[If[QQ[k*n+x^4+y^4-z^4],tab=Append[tab,m];Goto[aa]],
{x,0,m*(n^2/2)^(1/4)},{y,x,(k*n^2-x^4)^(1/4)},{z,0,((k*n+x^4+y^4)/2)^(1/4)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

A351341 Least nonnegative integer m such that n = x^4 + y^4 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

Original entry on oeis.org

0, 0, 0, 63, 3, 3, 4, 2, 2, 2, 4, 21, 37, 6, 1, 1, 0, 0, 4, 11, 7, 14, 5, 2, 2, 4, 8, 3, 3, 5, 1, 1, 0, 4, 4, 45, 5, 5, 11, 6, 6, 6, 32, 3, 7, 11, 3, 3, 6, 8, 8, 48, 13, 3, 3, 3, 6, 6, 31, 20, 93, 55, 3, 49, 33, 2, 2, 5, 5, 3, 3, 4, 2, 2, 2, 69, 17, 29, 11, 1, 1, 0, 0, 5, 61, 29, 8, 5, 2, 2, 4, 21, 29, 51, 6, 1, 1, 0, 4, 85, 13

Offset: 0

Zhi-Wei Sun, Feb 08 2022

nonn

Conjecture 1: Let k be 4 or 5. Then each integer can be written as x^k + y^k - (z^3 + w^3) with x,y,z,w nonnegative integers.
Two examples for k = 5: -4 = 58^5 + 76^5 - (775^3 + 1397^3) and 14 = 40^5 + 67^5 - (125^3 + 1132^3).
Conjecture 2: Let k be among 4, 5, 6 and 7. Then any integer can be written as x^k + y^k - (z^2 + w^2) with x,y,z,w nonnegative integers.
Examples for k = 6, 7: 170 = 9^6 + 15^6 - (2114^2 + 2730^2) and 469 = 7^7 + 8^7 - (1001^2 + 1385^2).
Conjecture 3: For any integer k > 3, there are no nonnegative integers x,y,z,w such that x^k + y^k - (z^k + w^k) = 3.
See also another similar conjecture in A351338.

			a(60) = 93 with 60 = 25^4 + 27^4 - (49^3 + 93^3).
a(527) = 527 with 527 = 29^4 + 110^4 - (91^3 + 527^3).
a(2198) = 1704 with 2198 = 85^4 + 304^4 - (1539^3 + 1704^3).
a(4843) = 1965 with 4843 = 142^4 + 338^4 - (1804^3 + 1965^3).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000578, A000583, A000584, A001014, A001015, A001481, A004831, A004999, A351306, A351321, A351338.

QQ[n_]:=IntegerQ[n^(1/4)];
tab={};Do[m=0;Label[bb]; k=m^3;Do[If[QQ[n+k+x^3-y^4], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^3)/2)^(1/4)}];m=m+1;Goto[bb];Label[aa],{n, 0, 100}];Print[tab]

A347824 Number of ways to write n as x^4 + y^4 + (z^2 + 23*w^2)/16, where x,y,z,w are nonnegative integers with x <= y.

Original entry on oeis.org

1, 2, 3, 3, 3, 2, 2, 1, 2, 3, 3, 2, 2, 3, 3, 1, 3, 4, 6, 4, 4, 1, 1, 2, 4, 7, 6, 4, 5, 6, 2, 2, 5, 5, 4, 3, 4, 3, 4, 3, 6, 8, 3, 4, 4, 2, 2, 3, 8, 5, 6, 2, 6, 5, 5, 6, 7, 2, 3, 4, 2, 2, 2, 4, 7, 5, 4, 1, 5, 3, 4, 7, 4, 6, 5, 4, 2, 1, 5, 5, 7, 7, 7, 6, 5, 3, 5, 4, 7, 7, 5, 4, 2, 5, 11, 7, 6, 9, 11, 5, 5

Offset: 0

Zhi-Wei Sun, Jan 23 2022

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,....
This has been verified for n up to 2*10^6. See also A347827 for a further refinement.
It seems that a(n) = 1 only for n = 0, 7, 15, 21, 22, 67, 77, 137, 252, 291, 437, 471, 477, 597, 1161, 4692, 7107.
For m = 32, 48, we also conjecture that every n = 0,1,2,... can be written as x^4 + y^4 + (z^2 + 23*w^2)/m, where x,y,z,w are nonnegative integers.

			a(7) = 1 with 7 = 0^4 + 1^4 + (2^2 + 23*2^2)/16.
a(15) = 1 with 15 = 1^4 + 1^4 + (1^2 + 23*3^2)/16.
a(67) = 1 with 67 = 1^4 + 2^4 + (15^2 + 23*5^2)/16.
a(477) = 1 with 477 = 0^4 + 2^4 + (27^2 + 23*17^2)/16.
a(597) = 1 with 597 = 2^4 + 4^4 + (5^2 + 23*15^2)/16.
a(1161) = 1 with 1161 = 2^4 + 2^4 + (89^2 + 23*21^2)/16.
a(4692) = 1 with 4692 = 2^4 + 5^4 + (248^2 + 23*12^2)/16.
a(7107) = 1 with 7107 = 1^4 + 5^4 + (239^2 + 23*45^2)/16.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000290, A000583, A004831, A347562, A347827, A349942, A349956.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[16(n-x^4-y^4)-23z^2],r=r+1],{x,0,(n/2)^(1/4)},{y,x,(n-x^4)^(1/4)},{z,0,Sqrt[16(n-x^4-y^4)/23]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A175372 Number of integer pairs (x,y) satisfying x^4 + y^4 = n.

Original entry on oeis.org

1, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

A 4th-power variant of A004018 and A175362.

a(n) is nonzero when n appears in A004831. a(n) > 8 when n appears in A003824. - Mason Korb, Oct 06 2018

G. C. Greubel, Table of n, a(n) for n = 0..10000
M. Korb, What's the connection between theta series and the number of integer solutions on a curve? , Math StackExchange, July 2018.

Cf. A004018, A175362.

Cf. A003824, A004831 (where a(n) is nonzero).

m:=120; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*(&+[x^(j^4): j in [1..50]]))^2)); // G. C. Greubel, Oct 06 2018

seq(coeff(series((1+2*add(x^(j^4),j=1..n))^2,x,n+1), x, n), n = 0 .. 120); # Muniru A Asiru, Oct 07 2018

CoefficientList[Series[(1 + 2*Sum[x^(j^4), {j, 1, 100}])^2, {x, 0, 120}], x] (* G. C. Greubel, Oct 06 2018 *)

x='x+O('x^120); Vec((1+2*sum(j=1,50, x^(j^4)))^2) \\ G. C. Greubel, Oct 06 2018

G.f.: (1 + 2*Sum_{j>=1} x^(j^4))^2.

A216280 Number of nonnegative solutions to the equation x^4 + y^4 = n.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

V. Raman, Sep 03 2012

nonn

The first n with a(n) > 1 is 635318657 = 41 * 113 * 241 * 569, with a(635318657) = 2. Izadi, Khoshnam, & Nabardi show that for any n with a(n) > 1, the elliptic curve y^2 = x^3 - nx has rank at least 3. According to gp, y^2 = x^3 - 635318657x has analytic rank 4 (and first nonzero derivative around 35741.7839). - Charles R Greathouse IV, Jan 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16561
F. A. Izadi, F. Khoshnam, K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥4 (2012). arXiv:1202.5676 [math.NT], 2012.
F. A. Izadi, F. Khoshnam, K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥ 4, Math. J. Okayama Univ. 56 (2014), 51-63.

Cf. A025455, A025446, A000161, A025426, A216284.

Cf. A004831 (positions of nonzero terms).

Reap[For[n = 1, n <= 1000, n++, r = Reduce[0 <= x <= y && x^4 + y^4 == n, {x, y}, Integers]; sols = Which[r === False, 0, r[[0]] == And, 1, r[[0]] == Or, Length[r], True, Print[n, " ", r]]; If[sols != 0, Print[n, " ", sols, " ", r]]; Sow[sols]]][[2, 1]] (* Jean-François Alcover, Feb 22 2019 *)

a(n)=my(t=thue(thueinit('x^4+1,1),n)); sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1]) \\ Charles R Greathouse IV, Jan 12 2017

first(n)=my(T=thueinit('x^4+1,1),v=vector(n),t); for(k=1,n, t=thue(T,k); v[k]=sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1])); v \\ Charles R Greathouse IV, Jan 12 2017

Offset added by Charles R Greathouse IV, Jan 12 2017

A247099 Numbers which are the sum or difference of two fifth powers.

Original entry on oeis.org

0, 1, 2, 31, 32, 33, 64, 211, 242, 243, 244, 275, 486, 781, 992, 1023, 1024, 1025, 1056, 1267, 2048, 2101, 2882, 3093, 3124, 3125, 3126, 3157, 3368, 4149, 4651, 6250, 6752, 7533, 7744, 7775, 7776, 7777, 7808, 8019, 8800, 9031, 10901, 13682, 15552, 15783, 15961

Offset: 1

Charles R Greathouse IV, Nov 29 2014

nonn

			31 = 2^5 - 1^5, 32 = 2^5 + 0^5, 33 = 2^5 + 1^5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000584, A004842, A004831, A181124.

Union[Flatten[{Total[#],Abs[#[[2]]-#[[1]]]}&/@Tuples[Range[0,10]^5,2]]] (* Harvey P. Dale, Mar 30 2015 *)

T=thueinit('z^5+1);
is(n)=n==0 || #thue(T, n)>0

a(n) << n^(5/2). Can this be improved?

cp's OEIS Frontend

A000583 Fourth powers: a(n) = n^4.

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A047217 Numbers that are congruent to {0, 1, 2} mod 5.

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A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

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

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A351306 Least positive integer m such that m^4*n = u^4 + v^4 - (x^4 + y^4) for some nonnegative integers u,v,x,y with x^4 + y^4 <= m^4*n^2.

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A351341 Least nonnegative integer m such that n = x^4 + y^4 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

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A351306 Least positive integer m such that m^4n = u^4 + v^4 - (x^4 + y^4) for some nonnegative integers u,v,x,y with x^4 + y^4 <= m^4n^2.