A213909 -id:A213909 - cp's OEIS frontend

A213916 Sum of all odd numbers in Collatz (3x+1) trajectory of n.

1, 1, 9, 1, 6, 9, 54, 1, 63, 6, 47, 9, 19, 54, 132, 1, 36, 63, 95, 6, 22, 47, 117, 9, 120, 19, 20262, 54, 76, 132, 20194, 1, 153, 36, 94, 63, 91, 95, 450, 6, 20235, 22, 248, 47, 81, 117, 20163, 9, 140, 120, 204, 19, 59, 20262, 20426, 54, 305, 76, 411, 132, 178

Offset: 1

Jayanta Basu, Mar 05 2013

nonn

			a(5)=6 since Collatz trajectory of 5 is 5,16,8,4,2,1.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A033493, A213909.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Total[Select[Collatz[n], OddQ]], {n, 100}] (* T. D. Noe, Mar 05 2013 *)

a(n) = A033493(n) - A213909(n).

A277068 a(n) = gcd(s1, s2), where s1 is the sum of the odd numbers and s2 is the sum of the even numbers in the Collatz (3x+1)trajectory of n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 18, 1, 3, 2, 1, 1, 1, 2, 2, 1, 2, 21, 1, 6, 2, 1, 3, 1, 2, 1, 2, 6, 2, 4, 2, 1, 1, 4, 2, 3, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 1, 2, 12, 2, 1, 1, 2, 2, 2, 1, 4, 3, 4, 2, 2, 2, 1, 5, 1, 1, 4, 2, 2, 2, 3, 1, 7, 2, 1, 1, 2, 2, 6, 7, 1, 1, 2, 2, 8

Offset: 1

Michel Lagneau, Sep 28 2016

nonn

Statistics of a(n) for the first 10^6 terms:
+------+-----------------+------------+
|      | number of terms |            |
|      |    such that    |            |
|   n  | gcd(s1, s2) = n | percentage |
+------+-----------------+------------+
|    1 |     401614      |   40.16%   |
|    2 |     305471      |   30.54%   |
|    3 |      44381      |    4.44%   |
|    4 |      76228      |    7.62%   |
|    5 |      15966      |    1.60%   |
|    6 |      34514      |    3.45%   |
|    7 |       8969      |    0.90%   |
|    8 |      19156      |    1.92%   |
|    9 |       4941      |    0.49%   |
|   10 |      12212      |    1.22%   |
|   11 |       3316      |    0.33%   |
|   12 |       8234      |    0.82%   |
| > 12 |      64998      |    6.50%   |
+------+-----------------+------------+
It seems that the values of the third column oscillate infinitely when n tend towards infinity.
Records: 1, 6, 18, 21, 23, 93, 187, 560, 1730, 5098, 10552, 11060, 11657, 31072, 32468, 306770, 793906, 1956888, 3107101, 12210181, etc.; they appear at 1, 5, 7, 18, 133, 147, 186, 270, 839, 5090, 5244, 5488, 23255, 62132, 113624, 153341, 793842, 6849034, 9321240, 12210146, etc. - Robert G. Wilson v, Oct 03 2016

			a(5)=6 because the Collatz trajectory of 5 is 5 -> 16 -> 8 -> 4 -> 2 -> 1 => s1 = 5+1 = 6, s2 = 16+8+4+2 = 30, and gcd(6, 30) = 6.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, The first occurrence of a(n)

Cf. A213909, A213916, A271973.

nn:=10^7:
for n from 1 to 100 do:
  m:=n:s1:=0:s2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m,2)=0
     then
     s2:=s2+m:m:=m/2:
     else
     s1:=s1+m:m:=3*m+1:
    fi:
   od:
     x:=gcd(s1+1,s2): printf(`%d, `,x):
  od:

Collatz[n_] := NestWhileList[ If[ OddQ[#], 3#+1, #/2] &, n, # > 1 &]; f[n_] := Block[{c = Collatz@ n}, GCD[Plus @@ Select[c, OddQ], Plus @@ Select[c, EvenQ]]]; Array[f, 86] (* Robert G. Wilson v, Oct 03 2016 *)

a(n) = {my(se = 0); my(so = 0); while (n!=1, if (n % 2, so+=n; n = 3*n+1, se +=n; n = n/2);); gcd(se, so+1);} \\ Michel Marcus, Oct 03 2016

A213917 Difference between sum of all even and the sum of all odd numbers in Collatz (3x+1) trajectory of n.

Original entry on oeis.org

-1, 1, 31, 5, 24, 37, 180, 13, 213, 34, 165, 49, 81, 194, 430, 29, 142, 231, 329, 54, 104, 187, 399, 73, 418, 107, 60916, 222, 290, 460, 60716, 61, 535, 176, 352, 267, 353, 367, 1444, 94, 60861, 146, 842, 231, 335, 445, 60653, 121, 526, 468, 722, 159, 281

Offset: 1

Jayanta Basu, Mar 05 2013

sign

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A213909, A213916.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[c = Collatz[n]; Total[Select[c, EvenQ]] - Total[Select[c, OddQ]], {n, 100}] (* T. D. Noe, Mar 05 2013 *)

a(n) = A213909(n) - A213916(n).

A271973 Smallest number k such that gcd(s1, s2) = n, where s1 is the sum of the odd numbers and s2 is the sum of the even numbers in the Collatz (3x+1) trajectory of k.

Original entry on oeis.org

1, 10, 9, 30, 65, 5, 74, 86, 368, 135, 970, 50, 95, 101, 1045, 178, 793, 7, 214, 196, 18, 423, 133, 200, 2572, 629, 621, 358, 700, 451, 3167, 1924, 3611, 1926, 662, 510, 6688, 437, 1525, 5072, 3724, 3161, 1034, 240, 5848, 2487, 704, 442, 19120, 1230, 5138, 3524

Offset: 1

Michel Lagneau, Jul 13 2016

nonn

			a(6) = 5 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 => gcd(6,30) = 6.

Cf. A213909, A213916.

nn:=10^8:
for n from 1 to 60 do:
ii:=0:
  for k from 1 to nn while(ii=0) do:
   kk:=1:m:=k:T[kk]:=k: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:
       g:=gcd(s1,s2):
       if g=n
       then
       ii:=1:printf("%d %d \n",n,k):
       else fi:
    od:
   od:

Table[k = 1; While[n != GCD[Total@ Select[#, OddQ], Total@ Select[#, EvenQ]] &@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, k, # > 1 &], k++]; k, {n, 52}] (* Michael De Vlieger, Jul 13 2016 *)

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.

Original entry on oeis.org

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

Michel Lagneau, Jul 07 2016

nonn

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.

			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.

Cf. A213909, A213916.

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:

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 *)

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

A277336 Numbers n for which the sum of the odd members and the sum of the even members in the Collatz (3x+1) trajectory are both semiprime.

Original entry on oeis.org

6, 12, 24, 35, 61, 76, 96, 118, 146, 162, 230, 245, 338, 362, 384, 426, 444, 460, 472, 580, 584, 605, 642, 645, 664, 697, 718, 740, 790, 804, 812, 814, 830, 852, 877, 920, 926, 954, 979, 1098, 1178, 1192, 1216, 1332, 1334, 1406, 1415, 1446, 1452, 1454, 1459

Offset: 1

Michel Lagneau, Oct 09 2016

nonn

The corresponding pairs of semiprimes are (9, 46), (9, 58), (9, 82), (94, 446), (178, 838), (95, 538), (9, 226), (411, 1894), (20499, 82366), (259, 1366), (493, 2446), (362, 1942), ...

			6 is in the sequence because the Collatz trajectory is 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 => the sum of the odd members is 3 + 5 + 1 = 9 = 3*3 and the sum of the even members is 6 + 10 + 16 + 8 + 4 + 2 = 46 = 2*23.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A213909, A213916, A275866.

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[PrimeOmega[s1]==2&&PrimeOmega[s2]==2,Print[n]],{n,1,1500}]

is(n)=my(e,o=1); while(n>1, if(n%2, o+=n; n+=2*n+1, e+=n; n/=2)); isprime(e/2) && bigomega(o)==2 \\ Charles R Greathouse IV, Oct 09 2016

A275584 Primes p such that S_e(p-1)/S_o(p-1) is an integer, where S_e(x) is the sum of the even numbers and S_o(x) is the sum of the odd numbers in the Collatz iteration of x.

Original entry on oeis.org

2, 3, 5, 17, 257, 59393, 65537, 331777, 534529, 1299457

Offset: 1

Jaroslav Krizek, Aug 04 2016

Primes p such that A213909(p-1)/A213916(p-1) is an integer.
Primes of the form A274796 + 1.
Fermat primes (A019434) are terms. Also supersequence of A092506 (primes of the form 2^n+1).
Corresponding values of S_e/o(a(n)-1): 0, 2, 6, 30, 510, 1567, 131070, ...

			Prime 59393 is a term because S_e/o(59392) = A213909(59392)/A213916(59392) = 119092/76 = 1567.

Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture

Cf. A033493, A213909, A213916, A274796.

[n+1: n in [A274796(m)] | IsPrime(n+1)]

e:= [&+[not IsOdd(h) select h else 0: h in [k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]]: n in [1..1000]]; o:= [&+[IsOdd(h) select h else 0: h in [k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]]: n in [1..1000]]; [n+1: n in [1..1000] | IsPrime(n+1) and e[n] mod o[n] eq 0]

Select[Prime@ Range[10^5], IntegerQ[Divide @@ Map[Total, TakeDrop[#, LengthWhile[#, EvenQ]]]] &@ SortBy[#, OddQ] &@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, # - 1, # > 1 &] &] (* Michael De Vlieger, Oct 15 2018 *)

S_e/o(A092506(n)-1) = A033493(A092506(n))-1.

More terms from Michael De Vlieger, Oct 15 2018.

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A213916 Sum of all odd numbers in Collatz (3x+1) trajectory of n.

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A277068 a(n) = gcd(s1, s2), where s1 is the sum of the odd numbers and s2 is the sum of the even numbers in the Collatz (3x+1)trajectory of n.

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A213917 Difference between sum of all even and the sum of all odd numbers in Collatz (3x+1) trajectory of n.

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A271973 Smallest number k such that gcd(s1, s2) = n, where s1 is the sum of the odd numbers and s2 is the sum of the even numbers in the Collatz (3x+1) trajectory of k.

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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.

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A277336 Numbers n for which the sum of the odd members and the sum of the even members in the Collatz (3x+1) trajectory are both semiprime.

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A275584 Primes p such that S_e(p-1)/S_o(p-1) is an integer, where S_e(x) is the sum of the even numbers and S_o(x) is the sum of the odd numbers in the Collatz iteration of x.

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