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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Or numbers n such that A213909(n)/A213916(n) is 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 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 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}]

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: A229083 A194415 A293536 * A160967 A326173 A045591

Adjacent sequences:  A274793 A274794 A274795 * A274797 A274798 A274799

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jul 07 2016

STATUS

approved

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Last modified February 17 12:14 EST 2020. Contains 331996 sequences. (Running on oeis4.)