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A340680 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A292251(i) = A292251(j), for all i, j >= 1.

%I A340680 #5 Feb 13 2021 07:46:08
%S A340680 1,2,3,2,1,2,4,2,1,2,5,2,6,2,7,2,1,2,3,2,1,2,4,2,1,2,8,2,1,2,9,2,10,2,
%T A340680 3,2,10,2,11,2,1,2,3,2,1,2,12,2,1,2,3,2,10,2,11,2,1,2,5,2,1,2,13,2,10,
%U A340680 2,8,2,1,2,11,2,1,2,3,2,6,2,14,2,1,2,8,2,1,2,4,2,1,2,5,2,10,2,15,2,10,2,5,2,1,2,16,2,1
%N A340680 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A292251(i) = A292251(j), for all i, j >= 1.
%C A340680 Restricted growth sequence transform of the ordered pair [A007814(1+n), A292251(n)], where the first element is the 2-adic valuation of 1+n (i.e., the number of trailing 1-digits in the base-2 representation of n), and the latter element is the 3-adic valuation of A048673(n).
%C A340680 For all i, j: a(i) = a(j) => A341345(i) = A341345(j).
%H A340680 Antti Karttunen, <a href="/A340680/b340680.txt">Table of n, a(n) for n = 1..65537</a>
%H A340680 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A340680 a(2n) = 2.
%o A340680 (PARI)
%o A340680 up_to = 65537;
%o A340680 rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A340680 A007814(n) = valuation(n,2);
%o A340680 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A340680 A048673(n) = (A003961(n)+1)/2;
%o A340680 A292251(n) = valuation(A048673(n),3);
%o A340680 Aux340680(n) = [A007814(1+n), A292251(n)];
%o A340680 v340680 = rgs_transform(vector(up_to, n, Aux340680(n)));
%o A340680 A340680(n) = v340680[n];
%Y A340680 Cf. A007814, A048673, A292251, A341345.
%Y A340680 Cf. also A322026.
%K A340680 nonn
%O A340680 1,2
%A A340680 _Antti Karttunen_, Feb 11 2021