A203075 Write n as a sum of distinct terms from A203074; if there is more than one way, choose the smallest binary number.

0, 1, 10, 11, 101, 110, 111, 1010, 1011, 1101, 1110, 1111, 10001, 10010, 10011, 10101, 10110, 10111, 11010, 11011, 11101, 11110, 11111, 100111, 101010, 101011, 101101, 101110, 101111, 110001, 110010, 110011, 110101, 110110, 110111, 111010, 111011

Offset: 0

Author

Frank M Jackson and N. J. A. Sloane, Dec 28 2011

Keywords

• nonn

Comments

a(n) is a binary vector for selecting terms from the complete sequence, A203074 that when summed gives n. It uses a miserly algorithm that chooses the smallest binary vector if there are multiple solutions. Somewhat similar to, although different from, A014417 and A104326.

Examples

			5 can be written as 5, i.e., 1000, or as 3+2, i.e., 110, and we choose the smaller.
18 can be written as 17+1, i.e., 100001, or as 11+5+2, i.e., 11010, and again we choose the smaller.
		

Links

• Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Crossrefs

Cf. A203974, A203076, A014417, A104326.

Programs

Mathematica

nextprime[n_Integer] := (k=n+1;While[!PrimeQ[k], k++];k);aprime[m_Integer] := (If[m==0, 1, nextprime[2^(m-1)]]);seqtable[l_] := (stable=Table[aprime[j], {j, 0, l}];stable);inttable[p_] := (itable=Reverse[IntegerDigits[p, 2]];itable);h=1;otable={0};ttable={};While[h<100, (inttable[h];seqtable[Length[itable]-1];test=itable.stable;If[!MemberQ[ttable, test], AppendTo[otable, h], Null];AppendTo[ttable, test];h++)];IntegerString[otable, 2]

Formula

a(n) x A203074 = n, where x is the inner product and the binary vector a(n) is in ascending powers of 2 with infinite trailing zeros.