A263570 - OEIS

A263570

Smallest positive integer such that n iterations of A073846 are required to reach an even number.

2, 3, 17, 31, 163, 353, 721, 1185, 1981, 3363, 5777, 10039, 29579, 52737, 94705, 171147, 311101, 568431, 1043463, 1923619, 3559911, 6611675, 12319517, 23023727, 651267929, 1234823707, 2345409699, 4462239583, 8502848523, 16226083005, 31007327791, 59331187155

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

A number is considered to be its own zeroth iteration.

Is the sequence defined for all n? If so, are there infinitely many composite numbers? If not, are infinitely many a(n) defined?

From Hartmut F. W. Hoft, Apr 05 2016: (Start)

Numbers a(6)...a(11) and a(12)...a(23) each belong to iteration sequences that start with prime numbers 10039 and 23023727, respectively, while the other numbers in the sequences are composite.

For the entire iteration sequences and computation of the additional numbers for this sequence see A271363. (End)

For n>1, a(n) is the least integer k such that the repeated application of x -> A073846(x) strictly decreases exactly n times in a row. - Hugo Pfoertner and Michel Marcus, Mar 11 2021

LINKS

Martin Ehrenstein, Table of n, a(n) for n = 0..43

FORMULA

For n>0, a(n+1) >= A073898(b(a(n))), where b(m) is the smallest odd composite not smaller than m, equality always holds if a(n) is composite.

EXAMPLE

a(2)=17 because A073846(17) = 15, A073846(15) = 14; thus it took two steps whereas no smaller positive integer has this property.

MATHEMATICA

(* Since A073846(9)=9, search starts with 11 *)

c25000000 = Select[Range[25000000], CompositeQ];

a073846[n_] := c25000000[[Floor[n/2]]]

a073846Nest[n_] := Length[NestWhileList[a073846, n, OddQ]]

a263570[n_] := Module[{list={2, 3}, i, length}, For[i=11, i<=n, i+=2, length=a073846Nest[i]; If[Length[list]<length, AppendTo[list, i]]]; list]