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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.

%I A282409 #17 May 10 2021 02:36:31
%S A282409 3,9,10,12,13,22,23,40,42,73,88,90,92,93,114,115,118,119,144,148,149,
%T A282409 152,154,162,163,164,165,166,192,208,212,213,226,227,251,295,318,319,
%U A282409 350,351,576,592,596,597,608,616,618,625,640,642,643,648,650,652,653
%N 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.
%C A282409 Or numbers m such that A078719(m) and A006666(m) are both prime.
%C A282409 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),...
%H A282409 Charles R Greathouse IV, <a href="/A282409/b282409.txt">Table of n, a(n) for n = 1..10000</a>
%H A282409 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A282409 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.
%p A282409 nn:=10^6:
%p A282409 for n from 1 to 500 do:
%p A282409   m:=n:i1:=1:i2:=0:
%p A282409    for i from 1 to nn while(m<>1) do:
%p A282409     if irem(m,2)=0
%p A282409      then
%p A282409      m:=m/2:i2:=i2+1:
%p A282409      else
%p A282409      m:=3*m+1:i1:=i1+1:
%p A282409     fi:
%p A282409    od:
%p A282409      if isprime(i1) and isprime(i2)
%p A282409      then
%p A282409      printf(`%d, `,n):
%p A282409      else
%p A282409     fi:od:
%o A282409 (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
%o A282409 (Python)
%o A282409 from sympy import isprime
%o A282409 def a(n):
%o A282409     l=[n]
%o A282409     while True:
%o A282409         if n%2==0: n//=2
%o A282409         else: n = 3*n + 1
%o A282409         l.append(n)
%o A282409         if n<2: break
%o A282409     o=list(filter(lambda i: i%2==1, l))
%o A282409     e=list(filter(lambda i: i%2==0, l))
%o A282409     return [o, e]
%o A282409 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
%Y A282409 Cf. A078719, A006666, A006667.
%K A282409 nonn
%O A282409 1,1
%A A282409 _Michel Lagneau_, Feb 14 2017