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A369042 LCM-transform of the inverse of binary Gray code (A006068).

%I A369042 #9 Jan 13 2024 13:08:03
%S A369042 1,3,2,7,1,2,5,1,1,1,13,2,3,11,1,31,1,1,29,1,5,3,1,2,17,19,1,23,1,1,1,
%T A369042 1,1,1,61,1,1,59,1,1,7,1,1,1,1,1,53,2,1,1,1,1,1,1,37,47,1,1,1,1,41,43,
%U A369042 1,127,1,1,5,1,11,1,1,1,113,1,1,1,1,1,1,1,97,1,1,103,1,1,101,1,1,1,109,1,1,107,1,2,1
%N A369042 LCM-transform of the inverse of binary Gray code (A006068).
%C A369042 Inverse of Binary Gray code, A006068, is a permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A006068(n)) = A000523(n), from which it immediately follows that A006068 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A006068(n)), for n >= 1.
%H A369042 Antti Karttunen, <a href="/A369042/b369042.txt">Table of n, a(n) for n = 1..16384</a>
%H A369042 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A369042 a(n) = lcm {1..A006068(n)} / lcm {1..A006068(n-1)}.
%F A369042 a(n) = A014963(A006068(n)). [See comments.]
%o A369042 (PARI)
%o A369042 up_to = 65537; \\ Checked up to 2^17;
%o A369042 LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
%o A369042 A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
%o A369042 v369042 = LCMtransform(vector(up_to,i,A006068(i)));
%o A369042 A369042(n) = v369042[n];
%o A369042 A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
%Y A369042 Cf. A006068, A014963, A368900, A369041, A369043, A369044.
%K A369042 nonn
%O A369042 1,2
%A A369042 _Antti Karttunen_, Jan 12 2024