A274796 Numbers n such that s2/s1 is an integer, where s1 is the sum of the odd numbers and s2 is the sum of the even numbers in the Collatz (3x+1) iteration of n.

1, 2, 4, 5, 8, 16, 20, 32, 64, 80, 128, 186, 256, 320, 512, 704, 1024, 1280, 1344, 2048, 3808, 4096, 5090, 5120, 6464, 8192, 10152, 15904, 16384, 20480, 21760, 28672, 32768, 34640, 59392, 62132, 65536, 81920, 106496, 131072, 138880, 217824, 262144, 327680

Offset

1,2

Comments

Or numbers n such that A213909(n)/A213916(n) is an integer.

The powers of 2 are in the sequence because s1 = 1.

The corresponding integers s2/s1 are 0, 2, 6, 5, 14, 30, 10, 62, 126, 30, 254, 6, 510, 110, 1022, 34, 2046, 430, 126, 4094, 14, 8190, 6, 1710, 70, 16382, 14, 37, 32766, 6830, 510, 1066, 65534, 26, 1567,... The odd numbers are very rare: 5, 37, 1567,...

The numbers of the form 5*2^2m for m = 0,1,.. are in the sequence because s1 = 6, s2 = (5*(2^(2m+1)-2)+ 30) ==0 (mod 6) => s2/s1 is an integer.

Links

Table of n, a(n) for n=1..44.

Example

5 is in the sequence because the Collatz trajectory of 5 is 5 -> 16 -> 8 -> 4 -> 2 -> 1 with s1 = 5+1 = 6 and s2 = 16 + 8 + 4 + 2 = 30 => 30/6 = 5 is an integer.

Maple

T:=array(1..2000):U:=array(1..2000):nn:=350000:
for n from 1 to nn do:
  kk:=1:m:=n:T[kk]:=n:it:=0:
    for i from 1 to nn while(m<>1) do:
     if irem(m, 2)=0
      then
       m:=m/2:kk:=kk+1:T[kk]:=m:
      else
      m:=3*m+1:kk:=kk+1:T[kk]:=m:
     fi:
    od:
    s1:=0:s2:=0:
    for j from 1 to kk do:
    if irem(T[j], 2)=1
    then
    s1:=s1+T[j]:
    else s2:=s2+T[j]:
    fi:
    od:
    if s1<>0 and floor(s2/s1)=s2/s1
    then
    printf(`%d, `, n):else fi:
  od:

Mathematica

coll[n_]:=NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]; a:=Select[coll[n], OddQ[#]&]; b:=Select[coll[n], EvenQ[#]&]; Do[s1=Sum[a[[i]], {i, 1, Length[a]}]; s2=Sum[b[[j]], {j, 1, Length[b]}]; If[IntegerQ[s2/s1], Print[n]], {n, 1, 350000}]
s2s1Q[n_]:=Module[{coll=NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&], s1, s2}, s1=Total[ Select[ coll, OddQ]]; s2=Total[Select[coll, EvenQ]]; IntegerQ[s2/s1]]; Select[Range[330000], s2s1Q] (* Harvey P. Dale, Feb 26 2024 *)

Prog

(PARI) isok(n) = {if (n % 2, s1 = n; s2 = 0, s2 = n; s1 = 0); while (n != 1, if (n % 2, n = 3*n+1, n /= 2); if (n % 2, s1 += n, s2 +=n); ); s2 % s1 == 0; } \\ Michel Marcus, Jul 09 2016

Crossrefs

Cf. A213909, A213916.

Sequence in context: A194415 A364779 A293536 * A160967 A326173 A045591

Adjacent sequences: A274793 A274794 A274795 * A274797 A274798 A274799

Keyword

nonn

Author

Michel Lagneau, Jul 07 2016

Status

approved