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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.

%I A185101 #25 Feb 16 2025 08:33:13
%S A185101 0,1,10,100,101,1000,1001,10000,1100,10010,10100,100000,11000,1000000,
%T A185101 100100,1000010,101000,10000000,110000,100000000,1010000,100000010,
%U A185101 10001000,1000000000,1100000,1000000010,100010000,1100100,10100000,10000000000
%N 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.
%C A185101 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 A185101 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>.
%F A185101 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 A185101 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 A185101 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 A185101 Cf. A007924, A066352, A200947.
%K A185101 nonn
%O A185101 0,3
%A A185101 _Frank M Jackson_, Jan 23 2012
%E A185101 Name clarified by _Frank M Jackson_, Oct 08 2013