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A198681
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Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3k+1.
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3
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3, 9, 24, 27, 42, 48, 60, 66, 72, 81, 96, 102, 114, 120, 126, 138, 144, 159, 168, 174, 180, 192, 198, 213, 216, 231, 237, 243, 258, 264, 276, 282, 288, 300, 306, 321, 330, 336, 342, 354, 360, 375, 378, 393, 399, 408, 414, 429, 432, 447, 453, 465, 471, 477, 492, 498
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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Select[3Range[200], IntegerQ[(Total[IntegerDigits[#, 3]]-1)/3]&] (* Harvey P. Dale, Feb 05 2012 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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