A282409 Numbers n for which the number of odd members and the number of even members in the Collatz (3x+1) trajectory are both prime.

3, 9, 10, 12, 13, 22, 23, 40, 42, 73, 88, 90, 92, 93, 114, 115, 118, 119, 144, 148, 149, 152, 154, 162, 163, 164, 165, 166, 192, 208, 212, 213, 226, 227, 251, 295, 318, 319, 350, 351, 576, 592, 596, 597, 608, 616, 618, 625, 640, 642, 643, 648, 650, 652, 653

Offset: 1

Author

Michel Lagneau, Feb 14 2017

Keywords

• nonn

Comments

Or numbers m such that A078719(m) and A006666(m) are both prime.

The distinct pairs of primes in the order of appearance are: (3, 5), (7, 13), (2, 5), (3, 7), (5, 11), (2, 7), (43, 73), (5, 13), (11, 23), (7, 17), (41, 71), (3, 11), (23, 43), (19, 37),...

Examples

			13 is in the sequence because its Collatz trajectory is 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 which contains 3 odd members and 7 even members.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A078719, A006666, A006667.

Programs

Maple

nn:=10^6:
for n from 1 to 500 do:
  m:=n:i1:=1:i2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m,2)=0
     then
     m:=m/2:i2:=i2+1:
     else
     m:=3*m+1:i1:=i1+1:
    fi:
   od:
     if isprime(i1) and isprime(i2)
     then
     printf(`%d, `,n):
     else
    fi:od:

PARI

is(n)=my(e,o=1); while(n>1, n=if(n%2, o++; 3*n+1, e++; n/2)); isprime(e) && isprime(o) \\ Charles R Greathouse IV, Feb 14 2017

Python

from sympy import isprime
def a(n):
    l=[n]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        l.append(n)
        if n<2: break
    o=list(filter(lambda i: i%2==1, l))
    e=list(filter(lambda i: i%2==0, l))
    return [o, e]
print([n for n in range(2, 1001) if isprime(len(a(n)[0])) and isprime(len(a(n)[1]))]) # Indranil Ghosh, Apr 14 2017