A334198 -id:A334198 - 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.

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

Antti Karttunen, Jan 04 2019

nonn

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

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

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.

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

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

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

A358730 Positions of first appearances in A358729 (number of nodes minus node-height).

Original entry on oeis.org

1, 4, 8, 16, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049

Offset: 1

Gus Wiseman, Dec 01 2022

First differs from A334198 in having 13122 instead of 12005.
Node-height is the number of nodes in the longest path from root to leaf.
After initial terms, this appears to become A038754.

			The terms together with their corresponding rooted trees begin:
      1: o
      4: (oo)
      8: (ooo)
     16: (oooo)
     27: ((o)(o)(o))
     54: (o(o)(o)(o))
     81: ((o)(o)(o)(o))
    162: (o(o)(o)(o)(o))
    243: ((o)(o)(o)(o)(o))
    486: (o(o)(o)(o)(o)(o))
    729: ((o)(o)(o)(o)(o)(o))

Positions of first appearances in A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by nodes and leaves.
MG differences: A358580, A358724, A358726, A358729.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A080936, A206487, A209638, A316321, A358576, A358577, A358592, A358725, A358731.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rd=Table[Count[MGTree[n],_,{0,Infinity}]-(Depth[MGTree[n]]-1),{n,10000}];
Table[Position[rd,k][[1,1]],{k,Union[rd]}]

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.

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