%I A019568
%S A019568 2,3,7,12,16,24,31,39,47,44,60,71,79
%N A019568 a(n) = smallest k >= 1 such that {1^n, 2^n, 3^n, ..., k^n} can be partitioned 
               into two sets with equal sum.
%C A019568 a(n) is least integer k such that at least one signed sum of the first 
               k n-th powers equals zero.
%C A019568 Contribution from Pietro Majer (majer(AT)dm.unipi.it), Mar 14 2009: (Start)
%C A019568 a(n)<2^(n+1).
%C A019568 The partition of the set {k: 0=<k<2^(n+1)} into two sets A,B according
%C A019568 to the parity of the number of 1s in the binary expansion of k, has the
%C A019568 property that sum_{k in A}p(k) = sum_{k in B}p(k) for any polynomial 
               p
%C A019568 of degree <= n. Equivalently, if e(k) is the Thue-Morse
%C A019568 sequence A106400, then sum_{0=<k<2^m} e(k)p(k)=0 for any
%C A019568 polynomial p with deg(p)<m. (End)
%D A019568 Posting to sci.math Nov 11 1996 by fredh(AT)ix.netcom.com (Fred W. Helenius).
%F A019568 a(n) == 0 or 3 (mod 4) for n >= 1 - David W. Wilson, Oct 20 2005.
%e A019568 For n=1 and 2 we have: 1+2-3 = 0 (so a(1)=3), 1+4-9+16-25-36+49 = 0 (so 
               a(2)=7).
%e A019568 The sum of the ninth powers of 3 5 9 10 14 19 20 21 25 26 28 31 35 36 
               37 38 40 41 42 is half the sum of the ninth powers of 1..44, so a(9)=44 
               - Don Reble, Oct 21 2005.
%e A019568 Contribution from Pietro Majer (majer(AT)dm.unipi.it), Mar 14 2009: (Start)
%e A019568 Example: the signs (+--+-++--++-+--+) in (+0)-1-8+27-64+125+216-343=0 
               are
%e A019568 those of the expansion of Q(x):=(1-x)(1-x^2)(1-x^4)(1-x^8)=
%e A019568 =+1-x-x^2+x^3-..+x^15. Since (1-x)^4 divides Q(x), if S is the shift
%e A019568 operator on sequences, the operator Q(S) has the fourth discrete
%e A019568 difference (I-S)^4 as factor, hence annihilates the sequence of cubes. 
               (End)
%Y A019568 Sequence in context: A046480 A137767 A080140 this_sequence A128458 A066733 
               A049623
%Y A019568 Adjacent sequences: A019565 A019566 A019567 this_sequence A019569 A019570 
               A019571
%K A019568 nonn
%O A019568 0,1
%A A019568 Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A019568 More from Don Reble, Oct 21 2005
%E A019568 Definition simplified by Pietro Majer (majer(AT)dm.unipi.it), Mar 15 
               2009