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a(n) is the smallest k such that the Collatz sequence for k includes a record number of consecutive tripling steps.
1

%I #24 Apr 14 2022 19:22:33

%S 1,3,7,15,27,127,255,511,1023,1819,4095,4255,16383,32767,65535,77671,

%T 262143,459759,1048575,2097151,4194303,7456539,16777215,33554431,

%U 67108863,125687199,1073741823,2147483647,4294967295,8589934591,17179869183,20361326439,68719476735

%N a(n) is the smallest k such that the Collatz sequence for k includes a record number of consecutive tripling steps.

%C See A350369 for a description of "consecutive tripling steps."

%C Records for A350369, recorded by the Collatz sequence starting value.

%C Differs from A213215 in that repeated values are removed, i.e., if a gap in the number of consecutive tripling steps occurs, A213215 will report the starting value multiple times but this sequence will not. Example: The Collatz sequence for 15 has 4 tripling steps but the sequence for 27 has 6, so 27 is reported by A213215 for n=5 and n=6. This sequence only reports 27 once as having set a new record.

%C Differs from A222598 in that certain consecutive tripling step lengths will not be represented here when a gap in the record number of consecutive tripling steps occurs. Example: Since the consecutive tripling step record moves from 4 in the Collatz sequence for 15 to 6 in the Collatz sequence for 27, this sequence will not report the Collatz sequence for 159 with 5 consecutive tripling steps like A222598 does.

%H Kevin P. Thompson, <a href="/A350370/b350370.txt">Table of n, a(n) for n = 1..44</a>

%e a(5) = 27 since the Collatz sequence for 27 is the 5th sequence to set a record for the most consecutive tripling steps, i.e., A350369(27) = 6 is the first occurrence of 6 in A350369.

%t k=0;nmax=0;Do[While[t=0;max=0;NestWhileList[If[OddQ@#,t++;If[t>max,max=t];(3#+1)/2,t=0;#/2]&,++k,#!=1&];max<nmax];nmax=max+1;Print@k,30] (* _Giorgos Kalogeropoulos_, Jan 11 2022 *)

%Y Cf. A006370, A213215, A222598, A347270, A347668, A350369.

%K nonn

%O 1,2

%A _Kevin P. Thompson_, Dec 27 2021