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A323424 Number of cycles (mod n) under Collatz map.

%I A323424 #11 Jan 17 2019 15:56:21
%S A323424 1,1,2,1,2,2,3,2,2,2,3,2,3,2,3,2,2,2,3,2,2,2,3,2,2,2,4,2,2,2,3,2,2,2,
%T A323424 3,2,3,3,3,2,3,2,4,2,3,2,3,2,3,2,3,2,2,2,4,2,2,2,3,2,2,2,3,2,2,2,3,2,
%U A323424 3,2,3,2,2,2,3,3,2,2,3,2,2,2,4,2,2,2,3
%N A323424 Number of cycles (mod n) under Collatz map.
%C A323424 This sequence is likely to be unbounded.
%H A323424 Rémy Sigrist, <a href="/A323424/a323424.png">Illustration for n = 13</a>
%H A323424 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A323424 a(n) >= 2 for any n > 4 (as we have at least the cycles (0) and (1, 4, 2)).
%e A323424 The initial terms, alongside the corresponding cycles, are:
%e A323424   n   a(n)  cycles
%e A323424   --  ----  --------------------
%e A323424    1     1  (0)
%e A323424    2     1  (0)
%e A323424    3     2  (0), (1)
%e A323424    4     1  (0)
%e A323424    5     2  (0), (1, 4, 2)
%e A323424    6     2  (0), (1, 4, 2)
%e A323424    7     3  (0), (1, 4, 2), (3)
%e A323424    8     2  (0), (1, 4, 2)
%e A323424    9     2  (0), (1, 4, 2)
%e A323424   10     2  (0), (1, 4, 2)
%e A323424   11     3  (0), (1, 4, 2), (5)
%e A323424   12     2  (0), (1, 4, 2)
%e A323424   13     3  (0), (1, 4, 2), (3, 10, 5)
%e A323424   14     2  (0), (1, 4, 2)
%e A323424   15     3  (0), (1, 4, 2), (7)
%e A323424   16     2  (0), (1, 4, 2)
%e A323424   17     2  (0), (1, 4, 2)
%e A323424   18     2  (0), (1, 4, 2)
%e A323424   19     3  (0), (1, 4, 2), (9)
%e A323424   20     2  (0), (1, 4, 2)
%o A323424 (PARI) a(n, f = k -> if (k%2, 3*k+1, k/2)) = { my (c=0, s=0); for (k=0, n-1, if (!bittest(s, k), my (v=0, i=k); while (1, v += 2^i; i = f(i) % n; if (bittest(s, i), break, bittest(v, i), c++; break)); s += v)); return (c) }
%Y A323424 See A000374, A023135, A023153, A233521 for similar sequences.
%Y A323424 Cf. A006370.
%K A323424 nonn
%O A323424 1,3
%A A323424 _Rémy Sigrist_, Jan 14 2019