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A369044
LCM-transform of bijective bit reverse (A057889).
6
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, 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, 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
OFFSET
1,2
COMMENTS
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.
FORMULA
a(n) = lcm {1..A057889(n)} / lcm {1..A057889(n-1)}.
a(n) = A014963(A057889(n)). [See comments.]
For n >= 1, Product_{d|n} a(A057889(d)) = n. [Implied by above.]
PROG
(PARI)
up_to = 65537; \\ Checked up to 2^17;
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); };
A030101(n) = if(n<1, 0, subst(Polrev(binary(n)), x, 2));
A057889(n) = if(!n, n, A030101(n/(2^valuation(n, 2))) * (2^valuation(n, 2)));
v369044 = LCMtransform(vector(up_to, i, A057889(i)));
A369044(n) = v369044[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 12 2024
STATUS
approved