A230562 -id:A230562 - cp's OEIS frontend

A230563 Smallest number that is the sum of three positive n-th powers in at least two ways.

5, 27, 251, 2673, 1375298099, 160426514

Offset: 1

Jonathan Sondow, Oct 25 2013

a(7) > 10^26 (if it exists). - Donovan Johnson, Nov 22 2013

a(7) > 33055^7 ~ 4.31*10^31 (if it exists). Duncan Moore, Oct 07 2017

			5 = 1^1 + 1^1 + 3^1 = 1^1 + 2^1 + 2^1.
27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2.
251 = 1^3 + 5^3 + 5^3 = 2^3 + 3^3 + 6^3.
2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4.
1375298099 = 3^5 + 54^5 + 62^5 = 24^5 + 28^5 + 67^5.
160426514 = 3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, section 21.11.

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. The value for a(6) given in Table 5 is wrong.

Cf. A046093, A230477, A230561, A230562.

a(4) and a(5) corrected by Donovan Johnson, Oct 28 2013

Edited by N. J. A. Sloane, Apr 03 2021

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

A374696 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

354, 6834, 16578, 300834, 2147874, 3847554, 16408434, 13155858, 489597858, 677125218, 780595299, 2374692243, 803898018, 5645172978

Offset: 1

Ilya Gutkovskiy, Jul 17 2024

			a(2) = 6834 = 1^4 + 2^4 + 4^4 + 9^4 = 3^4 + 4^4 + 7^4 + 8^4.
a(3) = 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.

Cf. A025417, A025421, A230562, A374693.

a(9)-a(14) from Michael S. Branicky, Jul 21 2024

A374693 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

98, 6578, 811538, 5978882, 1289202642, 292965218, 779888018, 5745705602, 105760443698, 49511121842, 1872511131218, 281539574498, 17673688436978

Offset: 1

Ilya Gutkovskiy, Jul 17 2024

a(16) = 7865870969138. - Michael S. Branicky, Jul 23 2024

			a(2) = 6578 = 1^4 + 2^4 + 9^4 = 3^4 + 7^4 + 8^4.
a(3) = 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.

Cf. A025415, A025419, A230562, A374696.

a(9)-a(12) from Michael S. Branicky, Jul 22 2024

a(13) from Michael S. Branicky, Jul 23 2024

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A230563 Smallest number that is the sum of three positive n-th powers in at least two ways.

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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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A374696 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

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A374693 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

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Author

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