cp's OEIS Frontend

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.

%I A274796 #13 Jan 10 2025 12:10:37
%S A274796 1,2,4,5,8,16,20,32,64,80,128,186,256,320,512,704,1024,1280,1344,2048,
%T A274796 3808,4096,5090,5120,6464,8192,10152,15904,16384,20480,21760,28672,
%U A274796 32768,34640,59392,62132,65536,81920,106496,131072,138880,217824,262144,327680
%N 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.
%C A274796 Or numbers n such that A213909(n)/A213916(n) is an integer.
%C A274796 The powers of 2 are in the sequence because s1 = 1.
%C A274796 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,...
%C A274796 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.
%e A274796 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.
%p A274796 T:=array(1..2000):U:=array(1..2000):nn:=350000:
%p A274796 for n from 1 to nn do:
%p A274796   kk:=1:m:=n:T[kk]:=n:it:=0:
%p A274796     for i from 1 to nn while(m<>1) do:
%p A274796      if irem(m,2)=0
%p A274796       then
%p A274796        m:=m/2:kk:=kk+1:T[kk]:=m:
%p A274796       else
%p A274796       m:=3*m+1:kk:=kk+1:T[kk]:=m:
%p A274796      fi:
%p A274796     od:
%p A274796     s1:=0:s2:=0:
%p A274796     for j from 1 to kk do:
%p A274796     if irem(T[j],2)=1
%p A274796     then
%p A274796     s1:=s1+T[j]:
%p A274796     else s2:=s2+T[j]:
%p A274796     fi:
%p A274796     od:
%p A274796     if s1<>0 and floor(s2/s1)=s2/s1
%p A274796     then
%p A274796     printf(`%d, `,n):else fi:
%p A274796   od:
%t A274796 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}]
%t A274796 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 *)
%o A274796 (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
%Y A274796 Cf. A213909, A213916.
%K A274796 nonn
%O A274796 1,2
%A A274796 _Michel Lagneau_, Jul 07 2016