A366388 The number of edges minus the number of leafs in the rooted tree with Matula-Goebel number n.

0, 0, 1, 0, 2, 1, 1, 0, 2, 2, 3, 1, 2, 1, 3, 0, 2, 2, 1, 2, 2, 3, 3, 1, 4, 2, 3, 1, 3, 3, 4, 0, 4, 2, 3, 2, 2, 1, 3, 2, 3, 2, 2, 3, 4, 3, 4, 1, 2, 4, 3, 2, 1, 3, 5, 1, 2, 3, 3, 3, 3, 4, 3, 0, 4, 4, 2, 2, 4, 3, 3, 2, 3, 2, 5, 1, 4, 3, 4, 2, 4, 3, 4, 2, 4, 2, 4, 3, 2, 4, 3, 3, 5, 4, 3, 1, 5, 2, 5, 4, 3, 3, 4, 2, 4

Offset: 1

Author

Antti Karttunen, Oct 23 2023

Keywords

• nonn

Comments

Number of iterations of A366385 needed to reach the nearest power of 2.

Examples

			See illustrations in A061773.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to Matula-Goebel numbers
• Index entries for sequences related to prime indices in the factorization of n

Crossrefs

Cf. A000720, A006530, A052126, A061395, A061773, A366385.

Cf. A109129 (gives the exponent of the nearest power of 2 reached), A196050 (distance to the farthest power of 2, which is 1).

Cf. also A329697, A331410.

Programs

Mathematica

Array[-1 + Length@ NestWhileList[PrimePi[#2]*#1/#2 & @@ {#, FactorInteger[#][[-1, 1]]} &, #, ! IntegerQ@ Log2[#] &] &, 105] (* Michael De Vlieger, Oct 23 2023 *)

PARI

A366388(n) = if(n<=2, 0, if(isprime(n), 1+A366388(primepi(n)), my(f=factor(n)); (apply(A366388, f[, 1])~ * f[, 2])));

PARI

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A366385(n) = { my(gpf=A006530(n)); primepi(gpf)*(n/gpf); };
A366388(n) = if(n && !bitand(n,n-1),0,1+A366388(A366385(n)));

Formula

Totally additive with a(2) = 0, and for n > 1, a(prime(n)) = 1 + a(n).

a(n) = A196050(n) - A109129(n).

a(2n) = a(A000265(n)) = a(n).