A152111 An increasing basis of order 3. See Comments for full definition.

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

Offset

1,3

Comments

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.

Numbers of the form A033045(k), or 2*A033045(k), or 4*A033045(k). - R. J. Mathar, Sep 21 2009

There are 3*2^i - 1 terms up to 8^i. - David A. Corneth, Aug 02 2017

Links

David A. Corneth, Table of n, a(n) for n = 1..12287 (terms up to 8^12).

Index entries for sequences that are an additive basis, order 3.

Maple

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

Prog

(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

Crossrefs

Cf. A008932, A033045, A152112, A284116.

Sequence in context: A023898 A125853 A080025 * A316856 A324524 A325621

Adjacent sequences: A152108 A152109 A152110 * A152112 A152113 A152114

Keyword

nonn,easy

Author

David S. Newman, Mar 22 2009

Extensions

More terms from R. J. Mathar, Sep 21 2009

Status

approved