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A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.

%I A198681 #28 Feb 16 2025 08:33:16
%S A198681 3,9,24,27,42,48,60,66,72,81,96,102,114,120,126,138,144,159,168,174,
%T A198681 180,192,198,213,216,231,237,243,258,264,276,282,288,300,306,321,330,
%U A198681 336,342,354,360,375,378,393,399,408,414,429,432,447,453,465,471,477,492,498
%N A198681 Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.
%C A198681 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.
%H A198681 Michel Marcus, <a href="/A198681/b198681.txt">Table of n, a(n) for n = 1..1001</a> (after Harvey P. Dale).
%H A198681 Chris Bernhardt, <a href="http://www.jstor.org/stable/27643161">Evil twins alternate with odious twins</a>, Math. Mag. 82 (2009), pp. 57-62.
%H A198681 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prouhet-Tarry-EscottProblem.html">Prouhet-Tarry-Escott Problem</a>
%t A198681 Select[3Range[200],IntegerQ[(Total[IntegerDigits[#,3]]-1)/3]&] (* _Harvey P. Dale_, Feb 05 2012 *)
%Y A198681 Cf. A000069, A001969, A157971, A158704, A198680, A198682.
%K A198681 nonn,base
%O A198681 1,1
%A A198681 _John W. Layman_, Oct 28 2011
%E A198681 Offset corrected by _Michel Marcus_, Mar 02 2016