

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.


12



2, 3, 7, 12, 16, 24, 31, 39, 47, 44, 60, 71, 79, 79, 87
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OFFSET

0,1


COMMENTS

a(n) is least integer k such that at least one signed sum of the first k nth powers equals zero.
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 ThueMorse 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


REFERENCES

Posting to sci.math Nov 11 1996 by fredh(AT)ix.netcom.com (Fred W. Helenius).


LINKS

Table of n, a(n) for n=0..14.
Sean A. Irvine, Java program (github)
Pietro Majer, MathOverflow: Asymptotic growth of a certain integer sequence


FORMULA

a(n) == 0 or 3 (mod 4) for n >= 1  David W. Wilson, Oct 20 2005


EXAMPLE

For n=1 and 2 we have: 1+23 = 0 (so a(1)=3), 1+49+162536+49 = 0 (so a(2)=7).
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
Example: the signs (++++++++) in (+0)18+2764+125+216...+3375=0 are those of the expansion of Q(x):=(1x)(1x^2)(1x^4)(1x^8) = +1xx^2+x^3..+x^15. Since (1x)^4 divides Q(x), if S is the shift operator on sequences, the operator Q(S) has the fourth discrete difference (IS)^4 as factor, hence annihilates the sequence of cubes.  Pietro Majer, Mar 14 2009


MATHEMATICA

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 *)


CROSSREFS

Cf. A240070 (partitioned into 3 sets).
Sequence in context: A275374 A168249 A080140 * A128458 A066733 A049623
Adjacent sequences: A019565 A019566 A019567 * A019569 A019570 A019571


KEYWORD

nonn,more


AUTHOR

Robert G. Wilson v


EXTENSIONS

More terms from Don Reble, Oct 21 2005
Definition simplified by Pietro Majer, Mar 15 2009
a(13)a(14) from Sean A. Irvine, Mar 27 2019


STATUS

approved



