login
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.
3
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 28 2016
STATUS
approved