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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
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.
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
Sequence in context: A194415 A364779 A293536 * A160967 A326173 A045591
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jul 07 2016
STATUS
approved