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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jan 12 2024

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

Cf. A014963, A054429, A368900, A369041, A369042, A369043, A369044.

Programs

PARI

up_to = 65537;
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); };
A054429(n) = ((3<<#binary(n\2))-n-1);
v369045 = LCMtransform(vector(up_to,i,A054429(i)));
A369045(n) = v369045[n];

PARI

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A054429(n) = ((3<<#binary(n\2))-n-1);
A369045(n) = A014963(A054429(n));

Formula

a(n) = lcm {1..A054429(n)} / lcm {1..A054429(n-1)}.

a(n) = A014963(A054429(n)). [See comments.]

For n >= 1, Product_{d|n} a(A054429(d)) = n. [Implied by above.]