A205598 - OEIS

%I #24 Jun 11 2023 23:29:23

%S 0,1,10,11,101,110,111,1010,1011,1101,1110,1111,10110,10111,11010,

%T 11011,11101,11110,11111,101011,101101,101110,101111,110110,110111,

%U 111010,111011,111101,111110,111111,1011110,1011111

%N The number n written using a minimizing algorithm in the base where the values of the places are 1 and primes.

%C Any nonnegative number can be written as a sum of distinct primes + e, where e is 0 or 1 (see A007924, which uses a greedy algorithm for writing n). However in this sequence a(n) is generated by using a minimizing algorithm that gives the smallest binary vector for select members from the sequence Q = (1 union primes) that when summed gives n. Without the minimizing condition there is ambiguity -- for example, 8 = 7+1 = 5+3 = 5+2+1 has three representations.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Complete_sequence">"Complete" sequence</a>. [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]

%F Let Q be the ordered sequence of (1 union primes), then a(n) x Q = n, where x is the inner product and the binary vector a(n) is in ascending powers of 2 with infinite trailing zeros.

%e 8 = 7+1 = 5+3 = 5+2+1, so a(8) = 1011.

%t aprime[n_] := If[n==0, 1, Prime[n]]; 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]

%Y Cf. A007924, A185101, A200947.

%K nonn

%O 0,3

%A _Frank M Jackson_, Feb 08 2012