OFFSET
1,1
COMMENTS
It appears that Sum[k^j, 0<=k<=2^n-1, k in A198680] = Sum[k^j, 0<=k<=2^n-1, k in A198681] = Sum[k^j, 0<=k<=2^n-1, k in A180682], for 0<=j<=n-1, which has been verified numerically in a number of cases. This is a generalization of Prouhet's Theorem (see the reference). To illustrate for j=3, we have Sum[k^3, 0<=k<=2^n-1, k in A198680] = {0,0,12636,1108809,94478400,7780827681,633724260624,51425722195929,4168024588857600,...}, Sum[k^3, 0<=k<=2^n-1, k in A198681] = {0,27,14580,1095687,94478400,7780827681,633724260624,51425722195929,4168024588857600,..., Sum[k^3, 0<=k<=2^n-1, k in A198682] = {0,216,7776,1121931,94478400,7780827681,633724260624,51425722195929,4168024588857600,...}, and it is seen that all three sums agree for n>=4=j+1.
LINKS
Michel Marcus, Table of n, a(n) for n = 1..1001 (after Harvey P. Dale).
Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62.
Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem
MATHEMATICA
Select[3Range[200], IntegerQ[(Total[IntegerDigits[#, 3]]-1)/3]&] (* Harvey P. Dale, Feb 05 2012 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
John W. Layman, Oct 28 2011
EXTENSIONS
Offset corrected by Michel Marcus, Mar 02 2016
STATUS
approved