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.
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
KEYWORD
nonn
AUTHOR
STATUS
approved