A230563
Smallest number that is the sum of three positive n-th powers in at least two ways.
Original entry on oeis.org
5, 27, 251, 2673, 1375298099, 160426514
Offset: 1
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.
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
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).
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(* 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 *)
A230477
Smallest number that is the sum of n positive n-th powers in >= n ways.
Original entry on oeis.org
1, 50, 5104, 236674, 9006349824, 82188309244
Offset: 1
1 = 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
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.
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.
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.
- A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover, NY, 1966, pp. 162-165, 290-291.
- R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.
Cf.
A146756 (smallest number that is the sum of n distinct positive n-th powers in exactly n ways),
A230561 (smallest number that is the sum of two positive n-th powers in >= n ways),
A091414 (smallest number that is the sum of n positive n-th powers in >= 2 ways).
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
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.
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
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.
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