A138753 Number of iterations of A138754 before reaching a number for the second time, when starting with n.

1, 4, 5, 3, 3, 5, 3, 8, 6, 4, 21, 17, 7, 7, 5, 5, 22, 24, 20, 18, 18, 16, 8, 6, 8, 6, 29, 23, 27, 23, 23, 21, 19, 19, 17, 21, 17, 15, 7, 7, 9, 60, 9, 9, 7, 30, 28, 26, 24, 26, 24, 24, 28, 24, 22, 20, 20, 22, 20, 18, 20, 18, 20, 18, 18, 16, 14, 12, 10, 12, 10, 61, 59, 55, 12, 10, 8, 31

Offset

1,2

Comments

This is a variation of A138752, giving the number of iterations of A138754 needed to get any number for the second time, while A138752 stops counting somehow arbitrarily at 1=primepi(2) or 4=primepi(7).

The map A138754 is a variation of the Collatz map where parity of the integers is replaced by p mod 3 for the primes.

For the Collatz map, we have the only fixed point 0=f(0) and all other numbers seem to end up in the cycle 1->4->2->1.

Here the only fixed point is 1=A138754(1) and all other numbers seem to end up in the cycle 4 -> 7 -> 5 -> 4 (corresponding to primes 7 -> 17 -> 11 -> 7).

Depending on which number among primepi({2,7,11,17}) is reached first, A138753(n) = A138752(n)+1,+3,+2 resp. +1. (A138752(n) is the length of the so-called GB-sequence starting with prime(n).)

Links

Paolo Xausa, Table of n, a(n) for n = 1..1459 (terms 1..500 from M. F. Hasler)

Georges Brougnard, Definition of GB-sequences.

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

a(n) = min { k>0 | A138754^k(n) = A138754^m(n) for some m>=0, m<k }.

If n is not in {1,4,5,7}, then a(n) = 1+a(A138754(n)).

Example

a(1)=1 since after 1 step we find 1 again.
a(4)=3 since 4 -> 7 -> 5 -> 4 under A138754.

Mathematica

A138754[n_]:=A138754[n]=With[{p=Prime[n]}, PrimePi[NextPrime[If[Mod[p, 3]==2, p/2, 2p]]]];
A138753[n_]:=Length[NestWhileList[A138754, n, UnsameQ, {1, 4}]]-1;
Array[A138753, 100] (* Paolo Xausa, Jul 28 2023*)

Prog

(PARI) A138753(n, c=0, t=[1, 1, 1]) = { until( t[c++%3+1]==n=A138754(n), t[c%3+1]=n); c}

Crossrefs

Cf. A124123, A006577, A171938, A138756 (record values/indices).

Cf. A138750, A138751, A138752, A138754.

Sequence in context: A057113 A060134 A200351 * A179410 A272874 A008962

Adjacent sequences: A138750 A138751 A138752 * A138754 A138755 A138756

Keyword

nonn

Author

M. F. Hasler, Apr 01 2008

Status

approved