A369041 LCM-transform of binary Gray code (A003188).

1, 3, 2, 1, 7, 5, 2, 1, 13, 1, 1, 1, 11, 3, 2, 1, 5, 3, 1, 1, 31, 29, 1, 1, 1, 23, 1, 1, 19, 17, 2, 1, 7, 1, 1, 1, 1, 53, 1, 1, 61, 1, 1, 1, 59, 1, 1, 1, 41, 43, 1, 1, 47, 1, 1, 1, 37, 1, 1, 1, 1, 1, 2, 1, 97, 1, 1, 1, 103, 101, 1, 1, 109, 1, 1, 1, 107, 1, 1, 1, 11, 1, 1, 1, 127, 5, 1, 1, 1, 1, 1, 1, 1, 113, 1, 1, 3

Offset: 1

Author

Antti Karttunen, Jan 12 2024

Keywords

• nonn

Comments

Binary Gray code, A003188, 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(A003188(n)) = A000523(n), from which it immediately follows that A003188 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A003188(n)), for n >= 1.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Scatterplot of a(n), n = 1..2^16.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000523, A003188, A014963, A368900, A369042, A369043, A369044.

Programs

Mathematica

nn = 120; a[1] = s[1] = 1; Do[s[n] = LCM[s[n - 1], BitXor[n, Floor[n/2]] ]; a[n] = s[n]/s[n - 1], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 24 2024 *)

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); };
A003188(n) = bitxor(n, n>>1);
v369041 = LCMtransform(vector(up_to,i,A003188(i)));
A369041(n) = v369041[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

Formula

a(n) = lcm {1..A003188(n)} / lcm {1..A003188(n-1)}.

a(n) = A014963(A003188(n)). [See comments.]