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
Examples
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.
Links
- 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.
Programs
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PARI
a(n)=my(v=vector(n, i, 1), N=n^(4/3), k); while(v[#v]
Formula
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
Comments