A225819 Consider the set of n-tuples such that the sum of cubes of the elements is equal to square of their sum; sequence gives largest element in all such tuples.

1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 42, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 115, 117, 120, 123, 126, 129, 132, 134, 137, 140, 143, 146, 149, 152, 155

Offset

1,2

Comments

Conjecture [Sen]: lim inf log_n a(n) >= 5/4.

Links

Balarka Sen, Table of n, a(n) for n = 1..500

John Mason, Generalising 'sums of cubes equal to squares of sums', The Mathematical Gazette 85:502 (2001), pp. 50-58.

W. R. Utz, The Diophantine Equation (x_1 + x_2 + ... + x_n)^2 = x_1^3 + x_2^3 + ... + x_n^3, Fibonacci Quarterly 15:1 (1977), pp. 14, 16. Part 1, part 2.

Formula

n <= a(n) <= n^(4/3), see A158649.

Example

Call an n-multiset with the sum of cubes of the elements equal to square of their sum an n-SCESS.
a(6) = 7 since the only 6-SCESS with the largest element >= 7 are (2, 4, 4, 5, 5, 7), (3, 3, 3, 3, 5, 7), (3, 4, 5, 5, 6, 7), (3, 5, 5, 5, 6, 7) and (4, 5, 5, 6, 6, 7) and none have an element larger than 7.
a(7) = 9 since the only 7-SCESS with the largest element >= 9 are (4, 4, 4, 5, 5, 5, 9), (4, 5, 5, 5, 6, 6, 9) and (6, 6, 6, 6, 6, 6, 9) and none have an element larger than 9.
a(8) = 10 since the only 8-SCESS with the largest element >= 10 are (2, 5, 5, 5, 5, 5, 6, 10), (2, 6, 6, 6, 6, 6, 6, 10), (3, 4, 5, 5, 5, 6, 7, 10), (3, 4, 5, 5, 6, 6, 7, 10), (3, 5, 5, 5, 6, 7, 7, 10), (3, 6, 6, 6, 7, 7, 7, 10), (4, 4, 4, 4, 4, 4, 6, 10), (4, 4, 4, 4, 5, 5, 7, 10), (4, 5, 5, 6, 6, 7, 8, 10), (5, 5, 5, 7, 7, 7, 8, 10) and (6, 6, 6, 6, 6, 6, 9, 10) and none have an element larger than 10.

Prog

(PARI) a(n)=my(v=vector(n, i, 1), N=n^(4/3), m=n); while(v[#v]<N, v[1]++; if(v[1]>N, for(i=2, N, if(v[i]<N, v[i]++; for(j=2, i-1, v[j]=v[i]); v[1]=max(v[i], m); break))); if(sum(i=1, n, v[i]^3)==sum(i=1, n, v[i])^2, m=max(m, v[1]))); m

Crossrefs

Cf. A158649, A225808.

Sequence in context: A189725 A248635 A171511 * A205805 A246372 A006254

Adjacent sequences: A225816 A225817 A225818 * A225820 A225821 A225822

Keyword

nonn

Author

Charles R Greathouse IV, Jimmy Zotos, and Balarka Sen, Jul 30 2013

Status

approved