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

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

Author

Antti Karttunen, Jan 12 2024

Keywords

• nonn

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000523, A014963, A057889, A368900, A369041, A369042, A369043, A369045.

Programs

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); };

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.]