|
| |
|
|
A332096
|
|
Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.
|
|
2
|
|
|
|
1, 1, 1, 3, 4, 2, 0, 1, 0, 1, 1, 7, 18, 28, 25, 0, 1, 8, 0, 7, 1, 1, 15, 64, 158, 271, 317, 126, 45, 17, 59, 14, 2, 15, 3, 0, 2, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 31, 210, 748, 1825, 3351, 4606, 3760, 398, 131, 299, 0, 318, 0, 8
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,4
|
|
|
COMMENTS
|
It is known (Sprague 1948, cf. A001661) that for any n, only a finite number of positive integers are not the sum of distinct positive n-th powers. Therefore each row is finite, their lengths are given by A332098.
The number of nonzero terms in row n is A332066(n).
The column of the first zero (exact solution m^n = Sum_{x in A} x^n) in each row is given by A030052, unless A030052(n) = A332066(n) + 1 = A332098(n) + 1.
|
|
|
LINKS
|
|
|
|
FORMULA
|
For m <= m* + 1, T(n,m) = m^n - Sum_{0 < x < m} x^n.
|
|
|
EXAMPLE
|
The table reads:
n\ m=1 2 3 4 5 6 7 8 9 10 11 12 13
----+--------------------------------------------------------------------------
2 | 1 3 4 2 0 1 0 1 (A332098(2) = 8.)
3 | 1 7 18 28 25 0 1 8 0 7 1
4 | 1 15 64 158 271 317 126 45 17 59 14 2 15 3 0 ...
5 | 1 31 210 748 1825 3351 4606 3760 398 131 299 0 318 0 8 ...
The first column is all ones (A000012), since {1..m-1} = {} for m = 1.
The second column is 2^n - 1 = A000225 \ {0}, since {1..m-1} = {1} for m = 2.
The third column is 3^n - 2^n - 1 = |A083321(n)| for n > 1.
|
|
|
PROG
|
(PARI) A332096(n, m, r=0)={if(r, (m<2||r<2^(n-1)) && return(r-1); my(E, t=1); while(m^n>=r, E=m--); E=abs(r-(m+!!E)^n); for(a=2, m, if(r<t+=a^n, t=a-1; break)); t>=m && return(min(E, r-t)); while(m>=t && E, E=min(self()(n, m-1, r-m^n), E); E && E=min(self()(n, m-=1, r), E)); E, m < n/log(2)+1.5, m^n-sum(x=1, m-1, x^n), self()(n, m-1, m^n))}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
