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A279119 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies gcd(i, j)=1.

%I A279119 #21 Dec 23 2016 22:01:19
%S A279119 0,0,0,1,0,2,0,3,1,4,0,5,0,6,3,7,0,8,0,9,4,10,0,11,1,12,6,13,0,14,0,
%T A279119 15,7,16,2,17,0,18,9,19,0,20,0,21,10,22,0,23,1,24,12,25,0,26,5,27,13,
%U A279119 28,0,29,0,30,15,31,6,32,0,33,16,34,0,35,0,36,18,37
%N A279119 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies gcd(i, j)=1.
%C A279119 Also, for n>1, a(n) equals the index of the class of n relatively to the algorithm described in A275246 (i.e., if a(n)=k, then n is of class P_k).
%C A279119 From _Rémy Sigrist_, Dec 21 2016: (Start)
%C A279119 For any prime p, the sequence b_p(n)=a(p*n) is a bijection from A000027 to A001477:
%C A279119 - b_p is injective: b_p(n)=b_p(m) implies p*n=p*m or gcd(p*n,p*m)=1; as p>1, gcd(p*n,p*m)>1, so p*n=p*m and n=m.
%C A279119 - b_p is surjective: by contradiction: let k be the least number such that b_p(n) never equals k; we have a set of k terms (i_1,...,i_k) such that b_p(i_j) = j-1 for any j between 1 and k; let l be the least value such that p^l > max({1, i_1,...,i_k}). Then, by definition of a, a(p^l)=k, and b_p(p^(l-1))=k, which is a contraction.
%C A279119 (End)
%H A279119 Rémy Sigrist, <a href="/A279119/b279119.txt">Table of n, a(n) for n = 1..10000</a>
%F A279119 a(2*n) = n-1 for any n>0.
%F A279119 a(n)=0 iff n belongs to A008578.
%F A279119 a(n)=1 iff n belongs to A001248.
%F A279119 a(n)=2 iff n belongs to A089581.
%F A279119 a(n)=3 iff n belongs to A275246.
%F A279119 a(n)=4 iff n belongs to A275248.
%F A279119 a(n)=5 iff n belongs to A275249.
%F A279119 a(n)=6 iff n belongs to A275251.
%F A279119 a(n)=7 iff n belongs to A275252.
%F A279119 a(n)=8 iff n belongs to A275253.
%o A279119 (PARI) g = vector(76, i, 1); for (n=1, #g, a = 0; while (gcd(g[a+1],n)>1, a++); g[a+1] *= n; print1 (a ", "))
%Y A279119 Cf. A008578, A001248, A089581, A275246, A275248, A275249, A275251, A275252, A275253.
%K A279119 nonn
%O A279119 1,6
%A A279119 _Rémy Sigrist_, Dec 06 2016