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A087666 Consider recurrence b(0) = n/3, b(k+1) = b(k)*floor(b(k)); a(n) is the least k such that b(k) is an integer, or -1 if no integer is ever reached. 10
0, 3, 4, 0, 1, 1, 0, 3, 2, 0, 3, 7, 0, 1, 1, 0, 2, 3, 0, 2, 2, 0, 1, 1, 0, 5, 5, 0, 5, 6, 0, 1, 1, 0, 9, 2, 0, 8, 3, 0, 1, 1, 0, 2, 5, 0, 2, 2, 0, 1, 1, 0, 3, 3, 0, 6, 3, 0, 1, 1, 0, 4, 2, 0, 6, 4, 0, 1, 1, 0, 2, 4, 0, 2, 2, 0, 1, 1, 0, 6, 4, 0, 3, 6, 0, 1, 1, 0, 3, 2, 0, 3, 4, 0, 1, 1, 0, 2, 3, 0, 2, 2, 0, 1, 1, 0, 4, 7, 0, 6, 6, 0, 1, 1, 0, 5, 2, 0, 4, 3, 0, 1, 1, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,2

COMMENTS

It is conjectured that an integer is always reached if the initial value n/3 is >= 2.

LINKS

Table of n, a(n) for n=6..130.

Benoit Cloitre, Graph of (sum(k=6,n,a(k))-2n)*n^(-1/2), pdf

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

FORMULA

a(n)=0 iff n == 0 (mod 3), a(n)==1 iff n == 1 or 2 (mod 3^2), a(n)=2 iff n == 14,22,25,26 (mod 3^3). In general a(n)=m iff n == x (mod 3^m) where x pertains to a set of 2^m distinct elements included in {1,2,...,(3^m)-1}. Conjecture: a(6) + a(7) + a(8) + ... + a(n) = 2n + O(sqrt(n)). - Benoit Cloitre, Sep 24 2012

MAPLE

# Gives right answer as long as answer is < M.

# This is better than the Mathematica or PARI programs.

M := 50; f := proc(n) local c, k, tn, tf; global M; k := n/3; c := 0; while whattype(k) <> 'integer' do tn := floor(k); tf := k-tn; tn := tn mod 3^50; k := tn*(tn+tf); c := c+1; od: c; end; # N. J. A. Sloane

MATHEMATICA

f[n_] := If[ Mod[3n, 3] == 0, 0, Length[ NestWhileList[ #1*Floor[ #1] &, n, !IntegerQ[ #2] &, 2]] - 1]; Table[f[n/3], {n, 6, 120}] (* Robert G. Wilson v *)

PROG

(PARI) a(n)=if(n<0, 0, c=n/3; x=0; while(frac(c)>0, c=c*floor(c); x++); x) \\ Benoit Cloitre, Sep 29 2003

(Python)

def A087666(n):

    c, x = 0, n

    a, b = divmod(x, 3)

    while b != 0:

        x *= a

        c += 1

        a, b = divmod(x, 3)

    return c # Chai Wah Wu, Mar 01 2021

CROSSREFS

Cf. A083863 (integer reached), A086336 and A087663 (records), A057016, A087710, A088706 (inverse).

Sequence in context: A247446 A131099 A098800 * A061353 A016653 A096088

Adjacent sequences:  A087663 A087664 A087665 * A087667 A087668 A087669

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Sep 27 2003

EXTENSIONS

More terms from Benoit Cloitre, Sep 29 2003

STATUS

approved

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Last modified April 21 02:10 EDT 2021. Contains 343143 sequences. (Running on oeis4.)