A046881 -id:A046881 - cp's OEIS frontend

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

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

A374418 a(n) is the smallest number which can be represented as the sum of 2 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

Original entry on oeis.org

7, 325, 87539319

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

A374421 a(n) is the smallest number which can be represented as the sum of 3 distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

8, 62, 1009, 6578, 1375298099, 160426514

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

a(7) > 10^20, if it is not -1. - Michael S. Branicky, Jul 09 2024

			a(5) = 1375298099 = 3^5 + 54^5 + 62^5 = 24^5 + 28^5 + 67^5.
a(6) = 160426514 = 3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6.

Cf. A046881, A374418, A374422, A374423, A374424, A374425.

A374422 a(n) is the smallest number which can be represented as the sum of 3 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

Original entry on oeis.org

9, 101, 5104, 811538

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
a(4) = 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.

Cf. A046881, A342893, A374418, A374421, A374423, A374424, A374425.

A374423 a(n) is the smallest number which can be represented as the sum of 3 distinct positive n-th powers in exactly 4 ways, or -1 if no such number exists.

Original entry on oeis.org

10, 161, 13896, 5978882

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

			a(3) = 13896 = 1^3 + 12^3 + 23^3 = 2^3 + 4^3 + 24^3 = 4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3.
a(4) = 5978882 = 3^4 + 40^4 + 43^4 = 8^4 + 37^4 + 45^4 = 15^4 + 32^4 + 47^4 = 23^4 + 25^4 + 48^4.

Cf. A046881, A374418, A374421, A374422, A374424, A374425.

A374424 a(n) is the smallest number which can be represented as the sum of 4 distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

12, 90, 1036, 6834, 4062500, 160426515, 2056364173794800

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

			a(5) = 4062500 = 1^5 + 14^5 + 16^5 + 19^5 = 5^5 + 10^5 + 15^5 + 20^5.
a(6) = 160426515 = 1^6 + 3^6 + 19^6 + 22^6 = 1^6 + 10^6 + 15^6 + 23^6.

Cf. A046881, A374418, A374421, A374422, A374423, A374425.

a(7) from Michael S. Branicky, Jul 09 2024

A374425 a(n) is the smallest number which can be represented as the sum of 4 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

Original entry on oeis.org

13, 78, 1521, 16578, 1479604544, 1885800643779

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

			a(4) = 16578 = 1^4 + 2^4 + 9^4 + 10^4 = 2^4 + 5^4 + 6^4 + 11^4 = 3^4 + 7^4 + 8^4 + 10^4.
a(5) = 1479604544 = 3^5 + 48^5 + 52^5 + 61^5 = 13^5 + 36^5 + 51^5 + 64^5 = 18^5 + 36^5 + 44^5 + 66^5.

Cf. A046881, A374418, A374421, A374422, A374423, A374424.

a(6) from Michael S. Branicky, Jul 09 2024

A088703 Numbers of form x^5 + y^5, x,y > 0 and x <> y.

Original entry on oeis.org

33, 244, 275, 1025, 1056, 1267, 3126, 3157, 3368, 4149, 7777, 7808, 8019, 8800, 10901, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 32800, 33011, 33792, 35893, 40544, 49575, 59050, 59081, 59292, 60073, 62174, 66825, 75856, 91817

Offset: 1

Cino Hilliard, Nov 22 2003

nonn

Up to n = 100000, no instances occur where n is the sum of two distinct 5th powers in two different ways. Conjecture: no number can be expressed as the sum of two 5th powers in more than one way: A046881.

			33 = 2^5 + 1^5, so 33 is in sequence. 64 = 2^5 + 2^5 is not.

Guy, Richard K., Unsolved Problems in Number Theory, 2nd Ed., Springer-Verlag(1994), pp. 140.

David A. Corneth, Table of n, a(n) for n = 1..10000
Wikipedia, Generalized Taxicab Numbers

Subsequence of A003347.

Cf. A088687 (4th powers), A088677 (6th powers), A046881 (bounds for double reps).

lst={};e=5;Do[Do[x=a^e;Do[y=b^e;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/e)],a+1,-1}],{a,Floor[n^(1/e)],1,-1}],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Union[#[[1]]^5+#[[2]]^5&/@Subsets[Range[10],{2}]] (* Harvey P. Dale, Apr 25 2012 *)

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 Ralf Stephan, Dec 30 2004

A016078 Smallest number that is sum of 2 positive n-th powers in 2 different ways.

Original entry on oeis.org

4, 50, 1729, 635318657

Offset: 1

Robert G. Wilson v, Dec 11 1999

If it exists, a(5) > 1.02*10^26 (see eqn. (27) at the Mathworld link). - Jon E. Schoenfield, Jan 05 2019

			4 = 1^1 + 3^1 = 2^1 + 2^1;
50 = 1^2 + 7^2 = 5^2 + 5^2,
1729 = 1^3 + 12^3 = 9^3 + 10^3;
635318657 = 59^4 + 158^4 = 133^4 + 134^4 = A018786(1).

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.
Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers

Cf. A046881, A230561.

(* This is just an empirical verification *) Do[max = 4 + n^4; Clear[cnt]; cnt[] = 0; smallest = Infinity;  Do[ cnt[an = x^n + y^n] += 1; If[cnt[an] == 2 && an < smallest, smallest = an], {x, 1, max}, {y, x, max}]; Print["a(", n, ") = ", smallest], {n, 1, 4}] (* _Jean-François Alcover, Aug 13 2013 *)

If A230561(n) exists, then a(n) <= A230561(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3. - Jonathan Sondow, Oct 24 2013 [Comment edited by N. J. A. Sloane, Apr 03 2021]

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.

cp's OEIS Frontend

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

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A374418 a(n) is the smallest number which can be represented as the sum of 2 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

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A374421 a(n) is the smallest number which can be represented as the sum of 3 distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

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A374422 a(n) is the smallest number which can be represented as the sum of 3 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

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A374423 a(n) is the smallest number which can be represented as the sum of 3 distinct positive n-th powers in exactly 4 ways, or -1 if no such number exists.

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A374424 a(n) is the smallest number which can be represented as the sum of 4 distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

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A374425 a(n) is the smallest number which can be represented as the sum of 4 distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

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A088703 Numbers of form x^5 + y^5, x,y > 0 and x <> y.

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A016078 Smallest number that is sum of 2 positive n-th powers in 2 different ways.

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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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