A227847 Number of tuples (x_1, x_2, ..., x_n) with 1 <= x_1 <= x_2 <= ... <= x_n such that Sum_{i=1..n} x_i^3 = (Sum_{i=1..n} x_i)^2 and Sum_{i=1..n-1} x_i^3 + (x_n-1)^3 + (x_n+1)^3 = (Sum_{i=1..n-1} x_i + 2x_n)^2.

0, 1, 1, 1, 2, 2, 2, 6, 10, 31, 77, 206, 568, 1704, 5037, 15554

Offset

1,5

Comments

An n-tuple meeting the first condition is called an n-SCESS ("sum of cubes equals square of sum").

In other words, a(n) is the number of tuples (x_1, x_2, ..., x_n) satisfying SCESS such that (x_1, x_2, ..., x_{n-1}, x_n - 1, x_n + 1) also satisfies SCESS. - Max Alekseyev, Mar 04 2025

x_1 + x_2 + ... + x_{n-1} = A152948(x_n). - Balarka Sen, Aug 01 2013

Links

Table of n, a(n) for n=1..16.

Edward Barbeau and Samer Seraj, Sum of cubes is square of sum, arXiv:1306.5257 [math.NT], 2013.

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

Formula

A001055(n) <= a(n) <= A158649(n). - Balarka Sen, Aug 01 2013

Example

a(3) = 1 since the only 3-SCESS is (1, 2, 3) for which the corresponding ordered tuple (1, 2, 2, 4) satisfy the SCESS property. (See Mason et al.)
a(5) = 2 since the only 5-SCESS are (1, 2, 2, 3, 5) and (3, 3, 3, 3, 6) for which the corresponding ordered tuples (1, 2, 2, 3, 4, 6) and (3, 3, 3, 3, 5, 7) satisfy the SCESS property.
a(8) = 6 since the only 8-SCESS are (1, 1, 2, 4, 5, 5, 5, 8), (1, 2, 2, 3, 4, 5, 6, 8), (2, 2, 4, 4, 6, 6, 6, 9), (2, 6, 6, 6, 6, 6, 6, 10), (3, 3, 3, 3, 5, 6, 7, 9) and (3, 5, 5, 5, 6, 7, 7, 10) for which the corresponding ordered tuples (1, 1, 2, 4, 5, 5, 5, 7, 9), (1, 2, 2, 3, 4, 5, 6, 7, 9), (2, 2, 4, 4, 6, 6, 6, 8, 10), (2, 6, 6, 6, 6, 6, 6, 9, 11), (3, 3, 3, 3, 5, 6, 7, 8, 10) and (3, 5, 5, 5, 6, 7, 7, 9, 11) satisfy the SCESS property.

Prog

(PARI) a(n)=my(v=vector(n, i, 1), N=n^(4/3), k); while(v[#v]<N, v[1]++; if(v[1]>N, for(i=2, N, if(v[i]<N, v[i]++; for(j=1, i-1, v[j]=v[i]); break))); if(sum(i=1, n, v[i]^3)==sum(i=1, n, v[i])^2 && sum(i=2, n, v[i]^3)+(v[1]-1)^3+(v[1]+1)^3==(sum(i=2, n, v[i])+2*v[1])^2, k++)); k /* Balarka Sen, Aug 01 2013 */

Crossrefs

Cf. A001055, A152948, A158649, A225819.

Sequence in context: A024945 A186507 A032306 * A058756 A237614 A257255

Adjacent sequences: A227844 A227845 A227846 * A227848 A227849 A227850

Keyword

more,nonn,changed

Author

Jimmy Zotos, Aug 01 2013

Extensions

a(11)-a(15) from Balarka Sen, Aug 01 2013

a(16) from Balarka Sen, Aug 11 2013

Definition corrected by Max Alekseyev, Mar 04 2025

Status

approved