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

1,8

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

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

Mathematica

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

Prog

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

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.

Sequence in context: A306863 A334108 A332813 * A334201 A257400 A328081

Adjacent sequences: A323074 A323075 A323076 * A323078 A323079 A323080

Keyword

nonn

Author

Antti Karttunen, Jan 04 2019

Status

approved