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A369045 LCM-transform of binary invert permutation (A054429).

%I A369045 #10 Jan 13 2024 13:07:47
%S A369045 1,3,2,7,1,5,2,1,1,13,1,11,1,3,2,31,1,29,1,3,1,5,1,23,1,1,1,19,1,17,2,
%T A369045 1,1,61,1,59,1,1,1,1,1,53,1,1,1,7,1,47,1,1,1,43,1,41,1,1,1,37,1,1,1,1,
%U A369045 2,127,1,5,1,1,1,11,1,1,1,1,1,1,1,113,1,1,1,109,1,107,1,1,1,103,1,101,1,1,1,97,1,1,1
%N A369045 LCM-transform of binary invert permutation (A054429).
%C A369045 Binary invert permutation, A054429, 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(A054429(n)) = A000523(n), from which it immediately follows that A054429 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A054429(n)), for n >= 1.
%H A369045 Antti Karttunen, <a href="/A369045/b369045.txt">Table of n, a(n) for n = 1..16384</a>
%H A369045 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A369045 a(n) = lcm {1..A054429(n)} / lcm {1..A054429(n-1)}.
%F A369045 a(n) = A014963(A054429(n)). [See comments.]
%F A369045 For n >= 1, Product_{d|n} a(A054429(d)) = n. [Implied by above.]
%o A369045 (PARI)
%o A369045 up_to = 65537;
%o A369045 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 A369045 A054429(n) = ((3<<#binary(n\2))-n-1);
%o A369045 v369045 = LCMtransform(vector(up_to,i,A054429(i)));
%o A369045 A369045(n) = v369045[n];
%o A369045 (PARI)
%o A369045 A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
%o A369045 A054429(n) = ((3<<#binary(n\2))-n-1);
%o A369045 A369045(n) = A014963(A054429(n));
%Y A369045 Cf. A014963, A054429, A368900, A369041, A369042, A369043, A369044.
%K A369045 nonn
%O A369045 1,2
%A A369045 _Antti Karttunen_, Jan 12 2024