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

1,1

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),...

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

Example

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.

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:

Prog

(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

Crossrefs

Cf. A078719, A006666, A006667.

Sequence in context: A024574 A119203 A115403 * A059012 A352317 A055063

Adjacent sequences: A282406 A282407 A282408 * A282410 A282411 A282412

Keyword

nonn

Author

Michel Lagneau, Feb 14 2017

Status

approved