A185101 - OEIS

%I #24 Feb 25 2023 12:48:28

%S 0,1,10,100,101,1000,1001,10000,1100,10010,10100,100000,11000,1000000,

%T 100100,1000010,101000,10000000,110000,100000000,1010000,100000010,

%U 10001000,1000000000,1100000,1000000010,100010000,1100100,10100000,10000000000

%N 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.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>.

%F 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.

%e 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.

%t 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}]

%Y Cf. A007924, A066352, A200947.

%K nonn

%O 0,3

%A _Frank M Jackson_, Jan 23 2012

%E Name clarified by _Frank M Jackson_, Oct 08 2013