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A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

%I A292244 #28 Apr 08 2022 07:05:31
%S A292244 0,0,1,0,0,3,0,2,5,0,0,1,0,0,1,12,6,7,14,0,1,0,4,1,8,10,3,0,0,21,24,0,
%T A292244 1,28,2,3,2,0,1,0,0,5,2,2,1,22,24,17,0,12,33,32,14,35,42,28,45,24,0,1,
%U A292244 16,2,11,48,0,59,0,8,3,0,2,5,0,16,1,4,20,3,6,6,7,8,0,1,56,0,3,0,42,5,0,48,5,0,0,1,14,2,65,64,56,49,44,4,49,64,6,57,0
%N A292244 Base-2 expansion of a(n) encodes the steps where multiples of 3 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.
%H A292244 Antti Karttunen, <a href="/A292244/b292244.txt">Table of n, a(n) for n = 1..16384</a>
%H A292244 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A292244 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A292244 a(n) = A291770(A292243(n)).
%F A292244 Other identities. For all n >= 1:
%F A292244 a(A048673(n)) = A292247(n).
%F A292244 a(n) + A292245(n) = A064216(n).
%F A292244 a(n) AND A292245(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).
%e A292244 For n = 3, the starting value is a multiple of three, after which follows A253889(3) = 1, the end point of iteration, which is not a multiple of three, thus a(3) = 1*(2^0) = 1.
%e A292244 For n = 8, the starting value is not a multiple of three, after which follows A253889(8) = 3, which is, thus a(8) = 0*(2^0) + 1*(2^1) = 2.
%e A292244 For n = 9, the starting value is a multiple of three, after which follows A253889(9) = 8 (which is not), while A253889(8) = 3 (which is), thus a(9) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.
%t A292244 f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1];g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n;Table[FromDigits[#, 2] &@ Map[Boole[Divisible[#, 3]] &,  Reverse@ NestWhileList[Floor@ g[Floor[f[#]/2]] &, n, # > 1 &]], {n, 109}] (* _Michael De Vlieger_, Sep 16 2017 *)
%o A292244 (Scheme) (define (A292244 n) (A291770 (A292243 n)))
%Y A292244 Cf. A048673, A064216, A253889, A291770, A292243, A292247.
%Y A292244 Cf. also A292245, A292246, and A292381, A292383, A292385, and A292590, A292591 for similarly constructed sequences, and also A292250.
%K A292244 nonn,base,look
%O A292244 1,6
%A A292244 _Antti Karttunen_, Sep 15 2017