A230477 -id:A230477 - 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

A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

Original entry on oeis.org

5104, 9729, 12104, 12221, 12384, 13896, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 40041, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 41966, 42056, 42687

Offset: 1

David W. Wilson

nonn

			a(1) = A230477(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3. - _Jonathan Sondow_, Oct 24 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A008917, A018787, A025331, A025397, A025402, A025407, A230477, A343968, A344239.

Select[ Range[ 50000], 2 < Length @ Cases[ PowersRepresentations[#, 3, 3], {?Positive, ?Positive, A008917%20by%20_Jonathan%20Sondow">?Positive}] &] (* adapted from Alcover's program for A008917 by _Jonathan Sondow, Oct 24 2013 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1,k,for(b=a,k,for(c=b,k,if(a^3+b^3+c^3==n,d++))));d
n=3;while(n<50000,if(is(n)>=3,print1(n,", "));n++) \\ Derek Orr, Aug 27 2015

A008917(n) < a(n) <= A025397(n). - Jonathan Sondow, Oct 24 2013

{n: A025456(n) >= 3}. - R. J. Mathar, Jun 15 2018

A146756 a(n) is the smallest number expressible as the sum of n distinct positive n-th powers in exactly n ways.

Original entry on oeis.org

1, 65, 5104, 300834, 9006349824, 82188309244

Offset: 1

Ian Rayson (Ian.Rayson(AT)studentmail.newcastle.edu.au), Nov 02 2008

It is not known if any terms exist in this sequence beyond n = 6.
Per email communication from Ian Rayson, the n-th powers for each sum must be distinct. If duplicate n-th powers were allowed, a(4) would be 236674 while the other terms would remain unchanged. - Donovan Johnson, Nov 08 2008
If duplicate n-th powers were allowed, a(2) would be 50. - Jonathan Sondow, Oct 23 2013

			a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
a(5) = 9006349824 = 8^5 + 34^5 + 62^5 + 68^5 + 92^5 = 8^5 + 41^5 + 47^5 + 79^5 + 89^5 = 12^5 + 18^5 + 72^5 + 78^5 + 84^5 = 21^5 + 34^5 + 43^5 + 74^5 + 92^5 = 24^5 + 42^5 + 48^5 + 54^5 + 96^5. - _Donovan Johnson_, Nov 08 2008
From _Michael S. Branicky_, Dec 21 2021: (Start)
a(6) = 82188309244 =  1^6 +  9^6 + 29^6 + 44^6 + 55^6 + 60^6,
                   =  2^6 + 12^6 + 25^6 + 51^6 + 53^6 + 59^6,
                   =  5^6 + 23^6 + 27^6 + 44^6 + 51^6 + 62^6,
                   = 10^6 + 16^6 + 41^6 + 45^6 + 51^6 + 61^6,
                   = 12^6 + 23^6 + 33^6 + 34^6 + 55^6 + 61^6,
                   = 15^6 + 23^6 + 31^6 + 36^6 + 53^6 + 62^6. (End)

Cf. A230477 (same except that the n-th powers need not be distinct and the number of ways is at least n, not necessarily exactly n). - Jonathan Sondow, Oct 23 2013

a(5) from Donovan Johnson, Nov 08 2008
Definition clarified by Jonathan Sondow, Oct 23 2013
a(6) from Michael S. Branicky, May 09 2021

A230561 Smallest number that is the sum of two positive n-th powers in >= n ways.

Original entry on oeis.org

2, 50, 87539319

Offset: 1

Jonathan Sondow, Oct 23 2013

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." Thus no one knows whether a(4) exists, which requires four such equal sums.

a(4) > 10^21 (if it exists). There is no number <= 10^21 that is the sum of two positive 4th powers in >= three ways. - Donovan Johnson, Jan 07 2014

			2 = 1^1 + 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.

Cf. A048610, A011541 for a(2), a(3).

Cf. also A016078, A230477.

a(n) >= A016078(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3.

A091414 Least number that is the sum of n positive n-th powers in at least 2 ways.

Original entry on oeis.org

50, 251, 259, 4097, 570947, 73310705, 647282661, 79327628290, 1077347903894, 1761813250036143, 2343908545594901

Offset: 2

Gabriel Cunningham (gcasey(AT)mit.edu), Mar 02 2004

From Donovan Johnson, Sep 14 2008: (Start)
a(11) = 2^11 + 2^11 + 2^11 + 2^11 + 8^11 + 10^11 + 10^11 + 15^11 + 22^11 + 22^11 + 22^11 = 3^11 + 5^11 + 5^11 + 5^11 + 6^11 + 9^11 + 11^11 + 12^11 + 17^11 + 20^11 + 24^11.
a(12) = 2^12 + 2^12 + 2^12 + 2^12 + 2^12 + 2^12 + 2^12 + 9^12 + 9^12 + 9^12 + 15^12 + 19^12 = 3^12 + 5^12 + 5^12 + 10^12 + 10^12 + 10^12 + 10^12 + 12^12 + 12^12 + 17^12 + 17^12 + 18^12.
a(13) > 876*10^15. a(14) > 799*10^15. a(15) > 115*10^16. (End)

			a(3) = 251 because 251 = 1^3 + 5^3 + 5^3 = 2^3 + 3^3 + 6^3 and it is the smallest number that can be represented two ways as the sum of three third powers.

a(2) = A048610(2), a(3) = A008917(1), a(4) = A185673(2). - Jonathan Sondow, Oct 24 2013

Cf. A230561, A230477.

a(n) <= A230477(n) for n > 1, with equality at least for n = 2 and inequality at least for n = 3, 4, 5. - Jonathan Sondow, Oct 24 2013

More terms from David Wasserman, Mar 09 2006
a(11)-a(12) from Donovan Johnson, Sep 14 2008
Definition shortened by Jonathan Sondow, Oct 24 2013

A219921 Numbers expressible as the sum of four nonnegative fourth-powers in four different ways.

Original entry on oeis.org

236674, 260658, 282018, 300834, 334818, 478338, 637794, 650034, 650658, 671778, 708483, 708834, 729938, 789378, 811538, 816578, 832274, 849954, 941859, 989043, 1042083, 1045539, 1099203, 1099458, 1102258, 1179378, 1243074, 1257954, 1283874, 1323234, 1334979

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 12 2013

nonn

A natural extension of the two-sets-of-two-cubes taxi-cab numbers (A001235).

a(4) is the first number which contains distinct fourth-powers in all four sets of four, and is therefore also A146756(4).

			a(1) = 236674 = 1^4+2^4+7^4+22^4 = 3^4+6^4+18^4+19^4 = 7^4+14^4+16^4+19^4 = 8^4+16^4+17^4+17^4.

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000
Christian N. K. Anderson, Decomposition of the first 1000 terms into four sets of four fourth powers.

Other sums of four fourth powers: A176197, A133526.
Cf. A146756, A230477.
Cf. A001235, A011541.

cp's OEIS Frontend

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

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A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

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A146756 a(n) is the smallest number expressible as the sum of n distinct positive n-th powers in exactly n ways.

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A230561 Smallest number that is the sum of two positive n-th powers in >= n ways.

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A091414 Least number that is the sum of n positive n-th powers in at least 2 ways.

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A219921 Numbers expressible as the sum of four nonnegative fourth-powers in four different ways.

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