

A061313


Minimal number of steps to get from 1 to n by (a) subtracting 1 or (b) multiplying by 2.


7



0, 1, 3, 2, 5, 4, 4, 3, 7, 6, 6, 5, 6, 5, 5, 4, 9, 8, 8, 7, 8, 7, 7, 6, 8, 7, 7, 6, 7, 6, 6, 5, 11, 10, 10, 9, 10, 9, 9, 8, 10, 9, 9, 8, 9, 8, 8, 7, 10, 9, 9, 8, 9, 8, 8, 7, 9, 8, 8, 7, 8, 7, 7, 6, 13, 12, 12, 11, 12, 11, 11, 10, 12, 11, 11, 10, 11, 10, 10, 9, 12, 11, 11, 10, 11, 10, 10, 9, 11, 10
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OFFSET

1,3


COMMENTS

Also number of steps to get from n to 1 by process of adding 1 if odd, or dividing by 2 if even.
It is straightforward to prove that the number n appears F(n) times in this sequence, where F(n) is the nth Fibonacci number (A000045).  Gary Gordon, May 31 2019
Conjecture: a(n)+2 is the sum of the terms of the Hirzebruch continued fraction for the SternBrocot tree fraction A007305(n)/A007306(n).  Andrey Zabolotskiy, Apr 17 2020


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divideandconquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divideandconquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index to sequences related to the complexity of n


FORMULA

a(2n) = a(n)+1; a(2n+1) = a(n+1)+2; a(1) = 0.
Is Sum_{k=1..n} a(k) asymptotic to C*n*log(n) where 3 > C > 2?  Benoit Cloitre, Aug 31 2002
G.f.: x/(1x) * Sum_{k>=0} (x^2^k + x^2^(k+1)/(1+x^2^k)).  Ralf Stephan, Jun 14 2003
a(n) = A080791(n1) + A029837(n).  Ralf Stephan, Jun 14 2003
a(n) = 2*A023416(n1) + A000120(n1) = A023416(A062880(n)) = A023416(A000695(n)) + 1.  Ralf Stephan, Jul 16 2003
a(n) = A119477(n)  1.  Philippe Deléham, Nov 03 2008


EXAMPLE

a(2) = 1 since 2 = 1*2, a(3) = 3 since 3 = 1*2*21, a(11) = 6 since 11 = (1*2*21)*2*21.


MATHEMATICA

f[n_] := Block[{c = 0, m = n}, While[m != 1, If[ EvenQ[m], While[ EvenQ[m], m = m/2; c++ ], m++; c++ ]]; Return[c]]; Table[f[n], {n, 1, 100}]


PROG

(PARI) a(n)=if(n<2, 0, s=n; c=1; while((s+s%2)/(2s%2)>1, s=(s+s%2)/(2s%2); c++); c)
(PARI) xpcount(n) = { p = 1; for(x=1, n, p1 = x; ct=0; while(p1>1, if(p1%2==0, p1/=2; ct++, p1 = p1*p+1; ct++) ); print1(ct, ", ") ) }
(Haskell)
a061313 n = fst $ until ((== 1) . snd) (\(u, v) > (u + 1, f v)) (0, n)
where f n = if r == 0 then n' else n + 1 where (n', r) = divMod n 2
 Reinhard Zumkeller, Sep 05 2015


CROSSREFS

Cf. A056792, A006577, A029837, A000045.
Cf. A007305, A007306, A008687.
Sequence in context: A076243 A140061 A292776 * A259846 A087669 A053087
Adjacent sequences: A061310 A061311 A061312 * A061314 A061315 A061316


KEYWORD

easy,nonn


AUTHOR

Henry Bottomley, Jun 06 2001


STATUS

approved



