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A152111
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An increasing basis of order 3. See Comments for full definition.
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3
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0, 1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 65, 72, 73, 128, 130, 144, 146, 256, 260, 288, 292, 512, 513, 520, 521, 576, 577, 584, 585, 1024, 1026, 1040, 1042, 1152, 1154, 1168, 1170, 2048, 2052, 2080, 2084, 2304, 2308, 2336, 2340, 4096, 4097, 4104, 4105, 4160
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OFFSET
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1,3
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COMMENTS
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Using the terminology of A008932, call a set A a basis of order h if every number can be written as the sum of h (not necessarily distinct) elements of A. Call a basis an increasing basis of order h if its elements are arranged in increasing order, a0 < a1 < a2 < ...
This sequence is constructed as follows: Take the union of the following three sets: (1) the set of all nonnegative numbers which can be written in base two as sums of powers, k, of 2, where k is congruent to 0 mod 3; (2) the set of all nonnegative numbers which can be written in base two as sums of powers, k, of 2, where k is congruent to 1 mod 3; (3) the set of all nonnegative numbers which can be written in base two as sums of powers, k, of 2, where k is congruent to 2 mod 3.
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LINKS
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MAPLE
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ismod3 := proc(n, m) b := convert(n, base, 2) ; for i from 1+((m+1) mod 3) to nops(b) by 3 do if op(i, b) <> 0 then RETURN(false) ; fi; od: for i from 1 + ((m+2) mod 3) to nops(b) by 3 do if op(i, b) <> 0 then RETURN(false) ; fi; od: true ; end: for n from 0 to 20700 do if ismod3(n, 0) or ismod3(n, 1) or ismod3(n, 2) then printf("%d, ", n); fi; od: # R. J. Mathar, Sep 21 2009
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PROG
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(PARI) upto(n) = {my(i = 1, r, res = List()); while(1, b = binary(i); r = sum(i=1, #b, 8^i*b[#b+1-i])>>3; if(r > n, break); listput(res, r); i+=2); q = #res; for(i=1, q, e = res[i] << 1; while(e <= n, listput(res, e); e=e<<1)); listput(res, 0); listsort(res); res} \\ David A. Corneth, Aug 02 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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