A073341 - OEIS

%I #21 Jan 31 2023 10:04:19

%S 3,2,3,8,1,1,3,2,2,3,2,2,1,1,7,4,4,2,4,3,1,1,2,4,2,8,4,3,1,1,6,4,3,2,

%T 5,4,1,1,5,2,2,3,2,2,1,1,4,5,6,2,3,5,1,1,2,3,2,4,3,6,1,1,7,8,3,2,4,5,

%U 1,1,3,2,2,3,2,2,1,1,7,3,4,2,7,6,1,1,2,5,2,5,5,3,1,1,3,3,3,2,10,3,1,1,4,2,2

%N Number of steps to reach an integer starting with (2n+1)/4 and iterating the map x -> x*ceiling(x).

%C We conjecture that an integer is always reached.

%C Is S(n) = Sum_{k=2..n} a(k) asymptotic to 3*n? S(n) = 3n for n = 69, 127, 166, 169, 189, 197, 327, 328, 360, 389, 404, 405, 419, 428, 497, 519, 520, 540, 541, 544, 547, 652, 668, 669, 676, 682, 683...

%C The sign of 3n-S(n) seems to change often: 3n-S(n) = 3, 4, 4, -1, 1, 3, 3, 4, 5, 5, 6, 7, 9, 11, 7, 6, 5, 6, 5, 5, 7, 9, 10, 9, 10, 5, 4, 4, 6, 8, 5, 4, 4, 5, 3, 2, 4, 6, 4, 5, 6, 6, 7, 8, 10, 12, 11, 9, 6, 7, 7, 5, 7, 9, 10, 10, 11, 10, 10, 7, 9, 11, 7, 2, 2, 3, 2, 0, 2, 4, 4, 5, 6, 6, 7, 8, 10, 12, 8, 8, 7, 8, 4, 1, 3, 5, 6, 4, 5, 3, 1, 1, 3, 5, 5, 5, 5, 6, -1, ... Is 3n-S(n) bounded? - _Benoit Cloitre_, Sep 05 2002

%H Robert Israel, <a href="/A073341/b073341.txt">Table of n, a(n) for n = 2..10000</a>

%H J. C. Lagarias and N. J. A. Sloane, Approximate squaring (<a href="http://neilsloane.com/doc/apsq.pdf">pdf</a>, <a href="http://neilsloane.com/doc/apsq.ps">ps</a>), Experimental Math., 13 (2004), 113-128.

%p g := proc(x) local M,t1,t2,t3; M := 4^100; t1 := ceil(x) mod M; t2 := x*t1; t3 := numer(t2) mod M; t3/denom(t2); end;

%p a := []; for n from 2 to 150 do t1 := (2*n+1)/4; for i from 1 to 100 do t1 := g(t1); if type(t1,`integer`) then break; fi; od: a := [op(a),i]; od: a;

%t a[n_] := Length @ NestWhileList[# Ceiling[#]&, (2n+1)/4, !IntegerQ[#]&] - 1;

%t Table[a[n], {n, 2, 100}] (* _Jean-François Alcover_, Jan 31 2023 *)

%o (PARI) a(n)=if(n<1,0,s=n/2+1/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c) \\ _Benoit Cloitre_, Sep 05 2002

%Y Cf. A073524, A074735, A085785, A085817, A085833.

%K nonn

%O 2,1

%A _N. J. A. Sloane_ and J. C. Lagarias, Sep 04 2002