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%I #29 Mar 28 2019 11:24:30
%S 2,3,7,12,16,24,31,39,47,44,60,71,79,79,87
%N 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 a(n) is least integer k such that at least one signed sum of the first k n-th powers equals zero.
%C a(n) < 2^(n+1). The partition of the set {k: 0 <= k < 2^(n+1)} into two sets A,B according to the parity of the number of 1s in the binary expansion of k, has the property that Sum_{k in A} p(k) = Sum_{k in B} p(k) for any polynomial p of degree <= n. Equivalently, if e(k) is the Thue-Morse sequence A106400, then Sum_{0 <= k < 2^m} e(k)p(k) = 0 for any polynomial p with deg(p) < m. - _Pietro Majer_, Mar 14 2009
%D Posting to sci.math Nov 11 1996 by fredh(AT)ix.netcom.com (Fred W. Helenius).
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a019/A019568.java">Java program</a> (github)
%H Pietro Majer, <a href="http://mathoverflow.net/questions/36995">MathOverflow: Asymptotic growth of a certain integer sequence</a>
%F a(n) == 0 or 3 (mod 4) for n >= 1 - _David W. Wilson_, Oct 20 2005
%e 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 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 Example: the signs (+--+-++--++-+--+) in (+0)-1-8+27-64+125+216-...+3375=0 are those of the expansion of Q(x):=(1-x)(1-x^2)(1-x^4)(1-x^8) = +1-x-x^2+x^3-..+x^15. Since (1-x)^4 divides Q(x), if S is the shift operator on sequences, the operator Q(S) has the fourth discrete difference (I-S)^4 as factor, hence annihilates the sequence of cubes. - _Pietro Majer_, Mar 14 2009
%t Table[k = 1; found = False; While[s = Range[k]^n; sm = Total[s]; If[EvenQ[sm], sm = sm/2; found = MemberQ[Total /@ Subsets[s], sm]]; ! found, k++]; k, {n, 0, 4}] (* _T. D. Noe_, Apr 01 2014 *)
%Y Cf. A240070 (partitioned into 3 sets).
%K nonn,more
%O 0,1
%A _Robert G. Wilson v_
%E More terms from _Don Reble_, Oct 21 2005
%E Definition simplified by _Pietro Majer_, Mar 15 2009
%E a(13)-a(14) from _Sean A. Irvine_, Mar 27 2019
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