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

1,5

Comments

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

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

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).

Example

See illustrations in A061773.

Mathematica

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

Prog

(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)));

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.

Sequence in context: A347386 A331410 A336928 * A114638 A123340 A360455

Adjacent sequences: A366385 A366386 A366387 * A366389 A366390 A366391

Keyword

nonn

Author

Antti Karttunen, Oct 23 2023

Status

approved