A003824 -id:A003824 - cp's OEIS frontend

A088867 Nonsquarefree elements of A003824, i.e., primitive solutions to a^4 + b^4 = c^4 + d^4 with nonsquarefree value on both sides of the equation.

680914892583617, 55683917506335026, 2056314197022256097, 3267700501872475297, 4544031582110882417, 10555434261160919777, 12361929340136667457, 23076050051029379057, 335875812638910622082

Offset: 1

Cino Hilliard, Nov 26 2003

nonn

Original definition was: Numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways that have at least one repeated factor.

Among the first 516 terms of A003824, there are 31 nonsquarefree terms. None of these are expressible in more than 2 ways as sum of two 4th powers. However, some of them, as 4544031582110882417, 12361929340136667457, 335875812638910622082, ..., have gcd(a,b) > 1, for one of the decompositions a^4 + b^4. - M. F. Hasler, Mar 05 2012

			a(1) = A003824(18) = 680914892583617 = 17^2*89*61657*429361 is the first nonsquarefree term of A003824. - _M. F. Hasler_, Mar 05 2012

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4

Cf. A003824, A088848, A088849.

\ begin a new session and type \r x4data.txt (evaluated Bernstein data) This will allow using %1 as the initial value. omegax4py42(n) = { for (i = 1, n, x = eval( Str("%", i) ); y=omega(x); y1 =bigomega(x); if(y<>y1,print1(x",")) ) }

select(A003824, t->!issquarefree(t))  \\ M. F. Hasler, Mar 05 2012

omega(n)<>bigomega(n) for n = a^4+b^4 = c^4+d^4 for distinct a, b, c, d. n=635318657, 3262811042, ..., 680914892583617, ..., 962608047985759418078417, ...

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

A343077 a(n) is the smallest number that is the sum of n positive 4th powers in two ways.

Original entry on oeis.org

635318657, 2673, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292

Offset: 2

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,4,2) in Alter's notation.

			a(2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.
a(3) = 2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 3, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003824, A046881, A342902, A343078, A343079, A343082, A343086.

a(n) = n + 240 for n >= 16.

A230562 Smallest number that is the sum of 2 positive 4th powers in >= n ways.

Original entry on oeis.org

0, 2, 635318657

Offset: 0

Jonathan Sondow, Oct 25 2013

Hardy and Wright say that a(3) is unknown.

Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums."

			0 = (empty sum).
2 = 1^4 + 1^4.
635318657 = 59^4 + 158^4 = 133^4 + 134^4.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, 2008; section 21.11.

Carlos Rivera, Puzzle 103: N = a^4 + b^4 = c^4 + d^4

Cf. A003824, A016078, A018786, A046881, A088687, A088848, A088849, A088867, A230561.

A088728 p such that p^4 + q^4 = r^4 + s^4 = a(n).

Original entry on oeis.org

158, 239, 292, 502, 542, 631, 1203, 1381, 2189, 2461, 2797, 2949, 3190, 3494, 3537, 4849, 4883, 5053, 6140, 6619, 6730, 6761, 7557, 7604, 8912, 8961, 9018, 9043, 9109, 9253, 9289, 9316, 9733, 10142, 10409, 10652, 11515, 12231, 12234, 12653, 13472

Offset: 1

Cino Hilliard, Nov 22 2003

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4

Cf. A003824, A088665.

A088848 Number of prime factors, without multiplicity, of numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways.

Original entry on oeis.org

4, 4, 4, 4, 3, 4, 4, 4, 6, 4, 5, 6, 4, 4, 7, 4, 7, 4, 3, 5, 6, 5, 6, 5, 6, 4, 5, 5, 6, 5, 4, 5, 4, 4, 6, 6, 6, 6, 6, 6, 5, 5, 6, 5, 6, 6, 6, 5, 7, 5, 6, 4, 5, 6, 6, 6, 5, 6, 5, 6, 4, 6, 4, 7, 6, 7, 5, 4, 5, 4, 5, 4, 6, 6, 5, 6, 5, 6, 5, 7, 4, 5, 6, 4, 6, 4, 6, 4, 5, 5, 9, 5, 5, 6, 6, 5, 3, 4, 5, 5

Offset: 1

Cino Hilliard, Nov 24 2003

			3262811042 = 2*113*2953*4889. Thus 4 is the first entry.

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4
Cino Hilliard, p,q,r,s and evaluation of the Bernstein data
Cino Hilliard, Evaluation of the Bernstein data only

Cf. A003824.

\ begin a new session and (back slash)r x4data.txt (evaluated Bernstein data) \ to the gp session. This will allow using %1 as the initial value. omegax4py42(n) = { for (i = 1, n, x = eval( Str("%", i) ); y=omega(x); print(y",") ) }

Omega(n) for n = a^4+b^4 = c^4+d^4 for distinct a, b, c, d. n=635318657, 3262811042, .., 960213785093149760746642, 962608047985759418078417

A088849 Number of prime factors, with multiplicity, of numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways.

Original entry on oeis.org

4, 4, 4, 4, 3, 4, 4, 4, 6, 4, 5, 6, 4, 4, 7, 5, 7, 4, 3, 5, 6, 5, 6, 5, 6, 4, 5, 5, 6, 5, 4, 5, 4, 4, 6, 6, 6, 6, 6, 6, 5, 5, 6, 5, 6, 6, 7, 5, 7, 5, 6, 4, 5, 6, 6, 6, 5, 6, 5, 6, 4, 6, 4, 7, 6, 7, 5, 4, 5, 4, 5, 4, 6, 6, 5, 6, 6, 6, 5, 7, 4, 5, 6, 4, 6, 5, 6, 4, 5, 8, 9, 5, 5, 6, 6, 5, 3, 5, 8, 5, 7, 5, 7, 6, 4

Offset: 1

Cino Hilliard, Nov 24 2003

			The 16th entry in the Bernstein Evaluation =
680914892583617 = 17*17*89*61657*429361 = 5 factors. 5 is the 16th entry in the sequence.

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4
Cino Hilliard, p,q,r,s and evaluation of the Bernstein data
Cino Hilliard, Evaluation of the Bernstein data only

Cf. A003824.

\ begin a new session and (back slash)r x4data.txt (evaluated Bernstein data) \ to the gp session. This will allow using %1 as the initial value. bigomegax4py42(n) = { for (i = 1, n, x = eval( Str("%", i) ); y=bigomega(x); print(y",") ) }

Bigomega(n) for n = a^4+b^4 = c^4+d^4 for distinct a, b, c, d. n=635318657, 3262811042, .., 680914892583617, .., 962608047985759418078417

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.

A088665 q such that p^4 + q^4 = r^4 + s^4 = a(n).

Original entry on oeis.org

59, 7, 193, 271, 103, 222, 76, 878, 1324, 1042, 248, 1034, 1577, 1623, 661, 3364, 2694, 604, 2027, 274, 2707, 498, 1259, 5181, 1657, 6262, 4903, 5098, 635, 1104, 1142, 173, 5452, 3401, 5277, 3779, 3644, 2903, 1525, 1149, 5121, 5526, 6470, 6496, 261, 581

Offset: 1

Cino Hilliard, Nov 23 2003

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4

Cf. A003824, A088728, A088867.

A008923 Euler's family of solutions to n = x^4 + y^4 = z^4 + w^4.

Original entry on oeis.org

90239171293339457, 43217672330080936976, 6822171645549542113537, 497455247066570553051152, 20549128177340906621890817, 24223393095189686902587392, 549140647573975773898200592

Offset: 1

N. J. A. Sloane

nonn

Mordell, Diophantine Equations, 1969, p. 90.

Cf. A003824.

Set x := a^7+a^5*b^2-2*a^3*b^4+3*a^2*b^5+a*b^6; y := a^6*b-3*a^5*b^2-2*a^4*b^3+a^2*b^5+b^7; z := a^7+a^5*b^2-2*a^3*b^4-3*a^2*b^5+a*b^6; w := a^6*b+3*a^5*b^2-2*a^4*b^3+a^2*b^5+b^7; then x^4+y^4=z^4+w^4.

cp's OEIS Frontend

A088867 Nonsquarefree elements of A003824, i.e., primitive solutions to a^4 + b^4 = c^4 + d^4 with nonsquarefree value on both sides of the equation.

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

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Mathematica

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A343077 a(n) is the smallest number that is the sum of n positive 4th powers in two ways.

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A230562 Smallest number that is the sum of 2 positive 4th powers in >= n ways.

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A088728 p such that p^4 + q^4 = r^4 + s^4 = a(n).

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A088848 Number of prime factors, without multiplicity, of numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways.

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A088849 Number of prime factors, with multiplicity, of numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways.

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

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Magma