A152111 - OEIS

%I #29 Jan 17 2024 09:15:04

%S 0,1,2,4,8,9,16,18,32,36,64,65,72,73,128,130,144,146,256,260,288,292,

%T 512,513,520,521,576,577,584,585,1024,1026,1040,1042,1152,1154,1168,

%U 1170,2048,2052,2080,2084,2304,2308,2336,2340,4096,4097,4104,4105,4160

%N An increasing basis of order 3. See Comments for full definition.

%C 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 < ...

%C 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.

%C Numbers of the form A033045(k), or 2*A033045(k), or 4*A033045(k). - _R. J. Mathar_, Sep 21 2009

%C There are 3*2^i - 1 terms up to 8^i. - _David A. Corneth_, Aug 02 2017

%H David A. Corneth, <a href="/A152111/b152111.txt">Table of n, a(n) for n = 1..12287</a> (terms up to 8^12).

%H <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.

%p 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

%o (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

%Y Cf. A008932, A033045, A152112, A284116.

%K nonn,easy

%O 1,3

%A _David S. Newman_, Mar 22 2009

%E More terms from _R. J. Mathar_, Sep 21 2009