The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A319385 Assuming the truth of the Collatz conjecture, let {m, f(m), f(f(m)), ..., 1} be the set where f is the Collatz function. The sequence lists the numbers m such that m/phi(m) + f(m)/phi(f(m)) + f(f(m))/phi(f(f(m)) + ... + 1/phi(1) is an integer, where phi is the Euler totient function A000010. 0
 1, 2, 4, 8, 16, 26, 32, 64, 128, 256, 512, 1024, 1664, 2048, 3392, 4096, 8192, 16384, 32768, 65536, 106496, 131072, 262144, 524288, 1048576, 2097152, 4194304, 6815744, 8388608, 16777216, 27918336, 33554432, 67108864, 134217728, 268435456, 436207616, 536870912 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The corresponding integers are 1, 3, 5, 7, 9, 21, 11, 13, 15, 17, 19, 21, 34, 23, 36, 25, 27, 29, 31, 33, 47, 35, 37, 39, 41, 43, 45, 60, 47, 49, 81, 51, 53, 55, 57, 73, 59, 61, 63, ... Conjecturally, it seems that all odd numbers are present, and the even numbers are rare: 34, 36, 60, ... We observe that the non-powers of 2 of the sequence: 26, 1664, 3392, 106496, 6815744, 27918336, 436207616, ... are of the form q*2^k with q in the set {13, 53, 213, ...}. LINKS EXAMPLE 26 is in the sequence because the Collatz trajectory is 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1, and 26/phi(26) + 13/phi(13) + 40/phi(40) + 20/phi(20) + 10/phi(10) + 5/phi(5) + 16/phi(16) + 8/phi(8) + 4/phi(4) + 2/phi(2) + 1/phi(1) = 26/12 + 13/12 + 40/16 + 20/8 + 10/4 + 5/4 + 16/8 + 8/4 + 4/2 + 2/1 + 1/1 = 21 is an integer. MAPLE with(numtheory):nn:=10^6: for n from 1 to 100000 do: T:=array(1..1000, [0\$1000]): it:=0:m:=n:k:=0:   for i from 1 to nn while(m<>1) do:     if irem(m, 2)=0      then      k:=k+1:T[k]:=m:m:=m/2:      else      k:=k+1:T[k]:=m:m:=3*m+1:     fi:    od:     k:=k+1:T[k]:=1:     s:=sum(‘T[i]/phi(T[i])’, ‘i’=1..k):      if s=floor(s)      then     printf (`%d %d \n`, n, s):    else fi:    od: CROSSREFS Cf. A000010, A000079, A006370, A070165. Sequence in context: A202274 A070789 A302588 * A180249 A060957 A322326 Adjacent sequences:  A319382 A319383 A319384 * A319386 A319387 A319388 KEYWORD nonn AUTHOR Michel Lagneau, Sep 18 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 8 10:59 EDT 2021. Contains 343666 sequences. (Running on oeis4.)