A071542 Number of steps to reach 0 starting with n and using the iterated process : x -> x - (number of 1's in binary representation of x).

0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..131072

Formula

a(0)=0, a(n) = 1 + A071542(n - A000120(n)). - Antti Karttunen, Oct 24 2012

It seems that a(n) ~ C n/log(n) asymptotically with C = 1.4... (n = 10^6 gives C = 1.469..., n = 10^7 gives C = 1.4614...).

Example

17 (= 10001 in binary) -> 15 (= 1111) -> 11 (= 1011) -> 8 (= 1000) -> 7 (= 111) -> 4 (= 100) -> 3 (= 11) -> 1 -> 0, hence a(17)=8.

Mathematica

Table[-1 + Length@ NestWhileList[# - DigitCount[#, 2, 1] &, n, # > 0 &], {n, 0, 75}] (* Michael De Vlieger, Jul 16 2017 *)

Prog

(PARI) for(n=1, 150, s=n; t=0; while(s!=0, t++; s=s-sum(i=1, length(binary(s)), component(binary(s), i))); if(s==0, print1(t, ", "); ); )
(PARI) a(n)=my(k); while(n, n-=hammingweight(n); k++); k \\ Charles R Greathouse IV, Oct 30 2012
(MIT/GNU Scheme)
;; with memoizing definec-macro:
(definec (A071542 n) (if (zero? n) n (+ 1 (A071542 (- n (A000120 n)))))) ;; Antti Karttunen, Oct 24 2012

Crossrefs

Cf. A000120, A011371, A071542, A213706, A213707, A213708, A218254.

A179016 gives the unique infinite sequence whose successive terms are related by this iterated process (in reverse order). Also, it seems that for n>=0, a(A213708(n)) = a(A179016(n+1)) = n.

A213709(n) = a((2^(n+1))-1) - a((2^n)-1).

Sequence in context: A274618 A176843 A238263 * A264810 A176841 A176814

Adjacent sequences: A071539 A071540 A071541 * A071543 A071544 A071545

Keyword

easy,nonn

Author

Benoit Cloitre, Jun 02 2002

Extensions

Starting offset changed to 0 with a(0) prepended as 0 by Antti Karttunen, Oct 24 2012

Status

approved