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A369044 LCM-transform of bijective bit reverse (A057889).

%I A369044 #10 Jan 13 2024 13:07:53
%S A369044 1,2,3,2,5,1,7,2,3,1,13,1,11,1,1,2,17,1,5,1,1,1,29,1,19,1,3,1,23,1,31,
%T A369044 2,1,1,7,1,41,1,1,1,37,1,53,1,1,1,61,1,1,1,1,1,43,1,59,1,1,1,1,1,47,1,
%U A369044 1,2,1,1,97,1,3,1,113,1,73,1,1,1,89,1,11,1,1,1,101,1,1,1,1,1,1,1,109,1,1,1,5,1,67
%N A369044 LCM-transform of bijective bit reverse (A057889).
%C A369044 Bijective bit reverse, A057889, 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(A057889(n)) = A000523(n), from which it immediately follows that A057889 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A057889(n)), for n >= 1.
%H A369044 Antti Karttunen, <a href="/A369044/b369044.txt">Table of n, a(n) for n = 1..16384</a>
%H A369044 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A369044 a(n) = lcm {1..A057889(n)} / lcm {1..A057889(n-1)}.
%F A369044 a(n) = A014963(A057889(n)). [See comments.]
%F A369044 For n >= 1, Product_{d|n} a(A057889(d)) = n. [Implied by above.]
%o A369044 (PARI)
%o A369044 up_to = 65537; \\ Checked up to 2^17;
%o A369044 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 A369044 A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
%o A369044 A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
%o A369044 v369044 = LCMtransform(vector(up_to,i,A057889(i)));
%o A369044 A369044(n) = v369044[n];
%o A369044 A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
%Y A369044 Cf. A000523, A014963, A057889, A368900, A369041, A369042, A369043, A369045.
%K A369044 nonn
%O A369044 1,2
%A A369044 _Antti Karttunen_, Jan 12 2024