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A369043 LCM-transform of Blue code (A193231).

%I A369043 #10 Jan 13 2024 13:07:57
%S A369043 1,3,2,5,2,1,7,1,1,1,13,1,11,3,2,17,2,1,19,1,1,23,1,1,31,29,1,3,1,1,5,
%T A369043 1,1,1,7,1,1,53,1,1,61,1,1,1,1,1,59,1,1,1,2,1,1,1,37,1,1,1,47,1,41,43,
%U A369043 1,1,1,1,1,1,3,83,1,1,1,89,1,1,1,1,1,1,1,71,1,1,2,1,67,1,1,1,73,1,79,1,1,1,103,101
%N A369043 LCM-transform of Blue code (A193231).
%C A369043 Blue code, A193231, is a self-inverse 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(A193231(n)) = A000523(n), from which it immediately follows that A193231 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A193231(n)), for n >= 1.
%H A369043 Antti Karttunen, <a href="/A369043/b369043.txt">Table of n, a(n) for n = 1..16384</a>
%H A369043 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A369043 a(n) = lcm {1..A193231(n)} / lcm {1..A193231(n-1)}.
%F A369043 a(n) = A014963(A193231(n)). [See comments.]
%F A369043 For n >= 1, Product_{d|n} a(A193231(d)) = n. [Implied by above.]
%o A369043 (PARI)
%o A369043 up_to = 65537; \\ Checked up to 2^17;
%o A369043 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 A369043 A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
%o A369043 v369043 = LCMtransform(vector(up_to,i,A193231(i)));
%o A369043 A369043(n) = v369043[n];
%o A369043 A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
%Y A369043 Cf. A014963, A193231, A368900, A369041, A369042, A369044, A369045.
%K A369043 nonn
%O A369043 1,2
%A A369043 _Antti Karttunen_, Jan 12 2024