A185101 The number n written using the minimum number of terms in the base where the values of the places are 1 and primes (noncomposites). For multiple solutions the smallest binary value is chosen.

0, 1, 10, 100, 101, 1000, 1001, 10000, 1100, 10010, 10100, 100000, 11000, 1000000, 100100, 1000010, 101000, 10000000, 110000, 100000000, 1010000, 100000010, 10001000, 1000000000, 1100000, 1000000010, 100010000, 1100100, 10100000, 10000000000

Offset

0,3

Comments

There are many ways of generating binary vectors a(n) for selecting noncomposites that when summed give n. A007924 uses the greedy algorithm. The above sequence uses the strong Goldbach conjecture that any integer is the sum of at most three distinct summands. It generates a(n) to select the minimum number of distinct noncomposites. Where there are multiple solutions, it chooses the smallest binary vector.

Links

Table of n, a(n) for n=0..29.

Eric Weisstein's World of Mathematics, Goldbach Conjecture.

Formula

For n, 1 to 6, a(n) is manually defined. For n prime, a(n) selects n. For n > 6 and n-2 prime, a(n) selects 2 and n-2. For n > 6 and even, use A082467(n/2) to give k, then a(n) selects n/2+k, n/2-k. For n>6 and odd, let m = (nextprime > n/3), then n-m is even and A082467((n-m)/2) gives k, a(n) selects m, (n-m)/2-k, (n-m)/2+k. If m = (n-m)/2+k, then m = nextprime(nextprime > n/3) and repeat.

Example

n=57 which is > 6 and odd, so m = (nextprime > 57/3) = 23 and n-m = 34 is even, thus A082467(17) = 6 and algorithm selects {23,11,23}. These are not distinct primes, so m = nextprime(nextprime > n/3) = 29 and A082467(14)=3, thus a(n) selects {29,11,17} as the binary vector 10010100000.

Mathematica

nextprime[j_] := Module[{k}, If[j==0, 1, (k=Floor[j]+1; While[!PrimeQ[k], k++]; k)]]; primetable[n_] := Module[{p, q}, Which[n==1, {0, 2, 0}, n==2, {1, 3, 0}, n==3, {1, 5, 0}, True, (p=n+1; q=2n-p; While[q>0&&!(PrimeQ[p]&&PrimeQ[q]), p++; q--]; {0, q, p})]]; fintable[m_] := Module[{temptable}, Which[m==0, {0, 0, 0}, m==1, {1, 0, 0}, PrimeQ[m], {0, m, 0}, PrimeQ[m-2]&&m>4, {0, 2, m-2}, EvenQ[m], primetable[m/2], True, (temptable=primetable[(m-nextprime[m/3])/2]; If[temptable[[3]]==nextprime[m/3], (temptable=primetable[(m-nextprime[nextprime[m/3]])/2]; temptable[[1]]=nextprime[nextprime[m/3]]), temptable[[1]]=nextprime[m/3]]; temptable)]]; decimal[t_] := Module[{temp2table, tempdecimal=0}, (temp2table=fintable[t]; If[temp2table[[1]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[1]]]]; If[temp2table[[2]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[2]]]]; If[temp2table[[3]]==0, Null, tempdecimal=tempdecimal+2^PrimePi[temp2table[[3]]]]; tempdecimal)]; Table[IntegerString[decimal[i], 2], {i, 0, 100}]

Crossrefs

Cf. A007924, A066352, A200947.

Sequence in context: A014417 A211027 A328072 * A007924 A115794 A105424

Adjacent sequences: A185098 A185099 A185100 * A185102 A185103 A185104

Keyword

nonn

Author

Frank M Jackson, Jan 23 2012

Extensions

Name clarified by Frank M Jackson, Oct 08 2013

Status

approved