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A205598 The number n written using a minimizing algorithm in the base where the values of the places are 1 and primes. 0
0, 1, 10, 11, 101, 110, 111, 1010, 1011, 1101, 1110, 1111, 10110, 10111, 11010, 11011, 11101, 11110, 11111, 101011, 101101, 101110, 101111, 110110, 110111, 111010, 111011, 111101, 111110, 111111, 1011110, 1011111 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Any nonnegative number can be written as a sum of distinct primes + e, where e is 0 or 1 (See A007924 that 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.

LINKS

Table of n, a(n) for n=0..31.

Wikipedia, Complete sequence.

FORMULA

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.

EXAMPLE

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

MATHEMATICA

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]

CROSSREFS

Cf. A007924, A185101, A200947

Sequence in context: A077813 A203075 A104326 * A037090 A171676 A118240

Adjacent sequences:  A205595 A205596 A205597 * A205599 A205600 A205601

KEYWORD

nonn

AUTHOR

Frank M Jackson, Feb 08 2012

STATUS

approved

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Last modified September 15 20:35 EDT 2014. Contains 246789 sequences.