This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344713 #17 Jul 04 2021 15:52:46 %S A344713 1,1,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,3,1,3,2,2, %T A344713 2,3,1,2,2,2,1,3,1,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,1,3,1,2,2,2,2,3,1,2, %U A344713 2,3,1,3,1,2,2,2,2,3,1,2,2,2,1,3,2,2,2,2,1,3,2,2,2,2,2,3,1,2,2,3,1,3,1,2,3,2,1,3,1,3 %N A344713 a(n) is the number of iterations needed for n to reach 0 under the mapping x -> A055212(x). %C A344713 Since x > A055212(x) for all positive integers x, and the smallest value of A055212(x) is 0, every trajectory under iteration of the mapping x -> A055212(x) will end at 0. %C A344713 If n = 1 or n is prime, then 0 will be reached in just one iteration of the mapping. Moreover, a(1), a(2), and a(3) form the only run of three consecutive 1's. All other 1's are isolated according to the prime numbers greater than 3. %C A344713 If n is a composite number, then its trajectory under the mapping consists of a first step n -> A055212(n) followed by a(A055212(n)) steps to reach 0. So, a(n) = a(A055212(n)) + 1. %e A344713 a(1) = 1, since 1 -> 0. %e A344713 a(p) = 1, since p -> 0 for any prime p. %e A344713 a(4) = 2, since 4 -> 1 -> 0. %e A344713 a(30) = 3, since 30 -> 4 -> 1 -> 0. %e A344713 a(1440) = 4, since 1440 -> 32 -> 4 -> 1 -> 0. %t A344713 f[0] = 0; f[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; a[n_] := -2 + Length @ FixedPointList[f, n]; Array[a, 100] (* _Amiram Eldar_, Jun 03 2021 *) %Y A344713 Cf. A000005, A036459, A055212. %K A344713 nonn %O A344713 1,4 %A A344713 _Timothy L. Tiffin_, May 26 2021