cp's OEIS Frontend

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

%I A323077 #30 Feb 28 2024 10:50:09
%S A323077 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,
%T A323077 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,
%U A323077 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
%N A323077 Number of iterations of map x -> (x - (largest divisor d < x)) needed to reach 1 or a prime, when starting at x = n.
%C A323077 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
%H A323077 Antti Karttunen, <a href="/A323077/b323077.txt">Table of n, a(n) for n = 1..12005</a>
%H A323077 Antti Karttunen, <a href="/A323077/a323077.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A323077 If A001222(n) <= 1 [when n is 1 or a prime], a(n) = 0, otherwise a(n) = 1 + a(A060681(n)).
%F A323077 a(n) <= A064097(n).
%F A323077 a(n) = A064097(n) - A334202(n) = A064097(A052126(n)). - _Antti Karttunen_, May 13 2020
%F A323077 a(A334198(n)) = n for all n >= 0. - _Antti Karttunen_, May 19 2020
%t A323077 Nest[Append[#1, If[PrimeOmega[#2] <= 1, 0, 1 + #1[[Max@ Differences@ Divisors[#2] ]] ]] & @@ {#, Length@ # + 1} &, {}, 105] (* _Michael De Vlieger_, May 26 2020 *)
%o A323077 (PARI)
%o A323077 A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
%o A323077 A323077(n) = if(1>=bigomega(n),0,1+A323077(A060681(n)));
%Y A323077 Cf. A006530, A032742, A052126, A060681, A064097, A323076, A334202, A334203.
%Y A323077 Cf. A334198 (positions of the records, also the first occurrence of each n).
%Y A323077 Differs from A334201 for the first time at n=169, where a(169) = 5, while A334201(169) = 6.
%K A323077 nonn
%O A323077 1,8
%A A323077 _Antti Karttunen_, Jan 04 2019