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.

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

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

Table of n, a(n) for n=1..37.

Index entries for sequences related to 3x+1 (or Collatz) problem

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: A348656 A354146 A354255 * A180249 A060957 A322326

Adjacent sequences: A319382 A319383 A319384 * A319386 A319387 A319388

Keyword

nonn

Author

Michel Lagneau, Sep 18 2018

Status

approved