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.

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

1,5

Comments

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

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Robert G. Wilson v, The first occurrence of a(n)

Example

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.

Maple

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:

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A213909, A213916, A271973.

Sequence in context: A281413 A049325 A250646 * A369904 A092371 A187552

Adjacent sequences: A277065 A277066 A277067 * A277069 A277070 A277071

Keyword

nonn

Author

Michel Lagneau, Sep 28 2016

Status

approved