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

Author

Michel Lagneau, Sep 28 2016

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A213909, A213916, A271973.

Programs

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 *)

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