A323077 Number of iterations of map x -> (x - (largest divisor d < x)) needed to reach 1 or a prime, when starting at x = n.

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 2, 0, 1, 2, 3, 0, 3, 0, 2, 2, 1, 0, 3, 3, 1, 4, 2, 0, 3, 0, 4, 2, 1, 3, 4, 0, 1, 2, 3, 0, 3, 0, 2, 4, 1, 0, 4, 4, 4, 2, 2, 0, 5, 3, 3, 2, 1, 0, 4, 0, 1, 4, 5, 3, 3, 0, 2, 2, 4, 0, 5, 0, 1, 5, 2, 4, 3, 0, 4, 6, 1, 0, 4, 3, 1, 2, 3, 0, 5, 4, 2, 2, 1, 3, 5, 0, 5, 4, 5, 0, 3, 0, 3, 5

Offset: 1

Author

Antti Karttunen, Jan 04 2019

Keywords

• nonn

Comments

When iteration is started from n, the first noncomposite reached is A006530(n), from which follows the new formula a(n) = A064097(A052126(n)) = A064097(n/A006530(n)), as A064097 is completely additive sequence. - Antti Karttunen, May 15 2020

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12005
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A006530, A032742, A052126, A060681, A064097, A323076, A334202, A334203.

Cf. A334198 (positions of the records, also the first occurrence of each n).

Differs from A334201 for the first time at n=169, where a(169) = 5, while A334201(169) = 6.

Programs

Mathematica

Nest[Append[#1, If[PrimeOmega[#2] <= 1, 0, 1 + #1[[Max@ Differences@ Divisors[#2] ]] ]] & @@ {#, Length@ # + 1} &, {}, 105] (* Michael De Vlieger, May 26 2020 *)

PARI

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323077(n) = if(1>=bigomega(n),0,1+A323077(A060681(n)));

Formula

If A001222(n) <= 1 [when n is 1 or a prime], a(n) = 0, otherwise a(n) = 1 + a(A060681(n)).

a(n) <= A064097(n).

a(n) = A064097(n) - A334202(n) = A064097(A052126(n)). - Antti Karttunen, May 13 2020

a(A334198(n)) = n for all n >= 0. - Antti Karttunen, May 19 2020