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A263570 Smallest positive integer such that n iterations of A073846 are required to reach an even number.

%I A263570 #38 Mar 22 2021 05:50:00
%S A263570 2,3,17,31,163,353,721,1185,1981,3363,5777,10039,29579,52737,94705,
%T A263570 171147,311101,568431,1043463,1923619,3559911,6611675,12319517,
%U A263570 23023727,651267929,1234823707,2345409699,4462239583,8502848523,16226083005,31007327791,59331187155
%N A263570 Smallest positive integer such that n iterations of A073846 are required to reach an even number.
%C A263570 A number is considered to be its own zeroth iteration.
%C A263570 Is the sequence defined for all n? If so, are there infinitely many composite numbers? If not, are infinitely many a(n) defined?
%C A263570 From _Hartmut F. W. Hoft_, Apr 05 2016: (Start)
%C A263570 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.
%C A263570 For the entire iteration sequences and computation of the additional numbers for this sequence see A271363. (End)
%C A263570 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
%H A263570 Martin Ehrenstein, <a href="/A263570/b263570.txt">Table of n, a(n) for n = 0..43</a>
%F A263570 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.
%e A263570 a(2)=17 because A073846(17) = 15, A073846(15) = 14; thus it took two steps whereas no smaller positive integer has this property.
%t A263570 (* Since A073846(9)=9, search starts with 11 *)
%t A263570 c25000000 = Select[Range[25000000], CompositeQ];
%t A263570 a073846[n_] := c25000000[[Floor[n/2]]]
%t A263570 a073846Nest[n_] := Length[NestWhileList[a073846, n, OddQ]]
%t A263570 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]
%t A263570 a263570[25000000] (* original sequence data *)
%t A263570 (* _Hartmut F. W. Hoft_, Apr 05 2016 *)
%Y A263570 Cf. A073846, A271363.
%K A263570 nonn
%O A263570 0,1
%A A263570 _Chayim Lowen_, Oct 21 2015
%E A263570 a(24)-a(31) from _Hartmut F. W. Hoft_, Apr 05 2016