

A258769


a(n) = Number of times the kth term is equal to k in the modified Collatz trajectory of n, when counting the initial term n as the 1st term: n, A014682(n), A014682(A014682(n)), ...


4



1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0
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OFFSET

1


COMMENTS

This sequence uses the definition given in A014682: if n is odd, n > (3n+1)/2 and if n is even, n > n/2.
2 occurs first at a(156) and 3 occurs first at a(153). Do all nonnegative numbers appear? See A258819.
"Number of fixed points in the modified Collatz trajectory of n."  This was the original name of the sequence, but is slightly misleading.  Antti Karttunen, Aug 18 2017


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem


EXAMPLE

For n = 6, the trajectory is given by T(6) = [6, 3, 5, 8, 4, 2, 1]. There are no values here such that T(6)[i] = i. So there are no fixed points, meaning a(6) = 0.
For n = 10, the trajectory is given by T(10) = [10, 5, 8, 4, 2, 1]. Here, the fourth term is 4, so there is a fixed point. Since there is only one, a(10) = 1.


PROG

(PARI) Tvect(n)=v=[n]; while(n!=1, if(n%2, k=(3*n+1)/2; v=concat(v, k); n=k); if(!(n%2), k=n/2; v=concat(v, k); n=k)); v
for(n=1, 200, d=Tvect(n); c=0; for(i=1, #d, if(d[i]==i, c++)); print1(c, ", "))
(Scheme)
(define (A258769 n) (if (= 1 n) n (let loop ((n n) (i 1) (s 0)) (if (= 1 n) s (loop (A014682 n) (+ 1 i) (+ s (if (= i n) 1 0)))))))
(define (A014682 n) (if (even? n) (/ n 2) (/ (+ n n n 1) 2)))
;; Antti Karttunen, Aug 18 2017


CROSSREFS

Cf. A014682, A070168, A258819, A258825 (variant where the indexing starts from k=0).
Sequence in context: A231367 A113428 A133101 * A266377 A266326 A185295
Adjacent sequences: A258766 A258767 A258768 * A258770 A258771 A258772


KEYWORD

nonn


AUTHOR

Derek Orr, Jun 09 2015


EXTENSIONS

Name changed by Antti Karttunen, Aug 18 2017


STATUS

approved



