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A198682
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Nonnegative multiples of 3 whose sum of base-3 digits are of the form 3*k+2.
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3
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6, 12, 18, 30, 36, 51, 54, 69, 75, 84, 90, 105, 108, 123, 129, 141, 147, 153, 162, 177, 183, 195, 201, 207, 219, 225, 240, 246, 252, 267, 270, 285, 291, 303, 309, 315, 324, 339, 345, 357, 363, 369, 381, 387, 402, 411, 417, 423, 435, 441, 456, 459, 474, 480, 486
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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.
For each m, the sequence contains exactly one of 9*m, 9*m+3, 9*m+6. - Robert Israel, Mar 04 2016
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LINKS
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MAPLE
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select(t -> convert(convert(t, base, 3), `+`) mod 3 = 2, [seq(3*i, i=1..1000)]); # Robert Israel, Mar 04 2016
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MATHEMATICA
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Select[Range[3, 498, 3], IntegerQ[(-2 + Plus@@IntegerDigits[#, 3])/3] &] (* Alonso del Arte, Nov 02 2011 *)
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PROG
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(PARI) isok(n) = !(n % 3) && !((vecsum(digits(n, 3)) - 2) % 3); \\ Michel Marcus, Mar 02 2016
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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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