A085276 -id:A085276 - cp's OEIS frontend

A072340 Number of steps to reach an integer starting with n/3 and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

0, 2, 6, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 1, 1, 0, 2, 4, 0, 3, 4, 0, 1, 1, 0, 22, 7, 0, 2, 5, 0, 1, 1, 0, 7, 2, 0, 4, 2, 0, 1, 1, 0, 2, 5, 0, 13, 9, 0, 1, 1, 0, 3, 3, 0, 2, 3, 0, 1, 1, 0, 3, 2, 0, 5, 2, 0, 1, 1, 0, 2, 3, 0, 8, 3, 0, 1, 1, 0, 5, 4, 0, 2, 4, 0, 1, 1, 0, 14, 2, 0, 3, 2, 0, 1, 1, 0, 2, 9, 0, 3, 9, 0, 1, 1

Offset: 3

N. J. A. Sloane and J. C. Lagarias, Sep 03 2002

nonn

We conjecture that an integer is always reached.

The occurrence of the first 1, 2, 3, 4 etc. is at the indices 7, 4, 13, 20, 10, 5, 29, 76, 50, 452, 244, 830, 49, 91, 319, 2639, 5753, 2215, 6151, 7148, 280, 28, 1783 - R. J. Mathar, Nov 25 2006

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

R. J. Mathar, Table of n, a(n) for n = 3..7147
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
N. J. A. Sloane, Seven Staggering Sequences.

Cf. A085276, A085285, A085286, A073524.

g := proc(x) local M,t1,t2,t3; M := 3^100; t1 := ceil(x) mod M; t2 := x*t1; t3 := numer(t2) mod M; t3/denom(t2); end;
f := proc(n) local t1,c; global g; if type(n, 'integer') then RETURN(0); fi; t1 := g(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := g(t1); od; RETURN(c); end;
[seq(f(n/3),n=3..120)]; # this gives the correct answer as long as the answer is < 99.

a[n_] := Module[{x = n/3, s = 0}, While[!IntegerQ[x], x *= Ceiling[x]; s++]; s]; Table[a[n], {n, 3, 107}] (* Jean-François Alcover, Jan 27 2019, from PARI *)

A072340(n)={ local(x,s) ; x=n/3 ; s=0 ; while( type(x)!="t_INT", x *= ceil(x) ; s++ ; ) ; return(s) ; } { for(n=3,10000, print(n," ",A072340(n)) ; ) ; } \\ R. J. Mathar, Nov 25 2006

A085285 Positions where records occur in A072340.

Original entry on oeis.org

3, 4, 5, 28, 1783, 7148, 273223, 398314, 1180939, 1751431, 10970993, 17545207, 20110862, 190270082, 10261454491

Offset: 1

N. J. A. Sloane, Aug 13 2003

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

Cf. A072340, A085276, A085286.

More terms from Vim Wenders, May 05 2008

A085286 Record values in A072340.

Original entry on oeis.org

0, 2, 6, 22, 23

Offset: 0

N. J. A. Sloane, Aug 13 2003

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

Cf. A072340, A085276, A085285.

A117596 Start with x=6/5; repeatedly apply the map x -> x*ceiling(x); sequence gives numerators of the resulting sequence of fractions.

Original entry on oeis.org

6, 12, 36, 288, 16704, 55808064, 622908012647232, 77602878444025201997703040704, 1204441348559630271252918141028336694332989128001036771264, 290135792424028156178425357986052529062710984863337179470336908191924417208517059859206222048920739921330978585792

Offset: 1

N. J. A. Sloane, Apr 07 2006

After 18 terms the fractions become integers, the first of which has 57735 digits.

			The sequence of fractions begins 6/5, 12/5, 36/5, 288/5, 16704/5, 55808064/5, 622908012647232/5, 77602878444025201997703040704/5, ... The first 17 denominators are 5, the rest are 1.

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

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

Cf. A072340, A085276.

f[x_] := x*Ceiling[x]; NestList[f, 6/5, 9] // Numerator (* Jean-François Alcover, Nov 18 2013 *)

A117620 Start with x=4/3; repeatedly apply the map x -> (x^2) ceiling(x); sequence gives numerators of the resulting sequence of fractions.

Original entry on oeis.org

4, 32, 4096, 285212672, 3536203627938199896064, 27735467127437590594631628902073909856749798039036448735232

Offset: 1

Jonathan Vos Post, Apr 07 2006

In this approximate cubing, does an iteration eventually yield an integer, after which denominators are 1? Fractions are 4/3, 32/9, 4096/81, 285212672/2187, 3536203627938199896064/1594323, 27735467127437590594631628902073909856749798039036448735232/2541865828329, 8393707510592229745861012598171776416393703955772365464679357805492895042198412632866136478758067686243059846017657263750451410617880163800261945260539460460740608/6461081889226673298932241.

a(9) has 1343 digits, and is too large for a b-file. - Robert Israel, Jun 15 2016

			a(4) = 285212672 because (4096/81)^2 * ceiling(4096/81) = (4096/81)^2 * ceiling(4096/81) = * ceiling(50.5679012) = (16777216/6561) * 51 = 285212672/2187.

Robert Israel, Table of n, a(n) for n = 1..8
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A072340, A085276, A117596.

x[1]:= 4/3:
for n from 1 to 9 do x[n+1]:= x[n]^2*ceil(x[n]) od:
seq(numer(x[i]),i=1..10); # Robert Israel, Jun 15 2016

Erroneous term removed by Giovanni Resta, Jun 15 2016

A117636 Start with x=4/3; repeatedly apply the map x -> x ceiling(x^2); sequence gives numerators of the resulting sequence of fractions.

Original entry on oeis.org

4, 8, 64, 9728, 920599396352, 780210979034070658749485424425566208

Offset: 1

Jonathan Vos Post, Apr 08 2006

In this approximate cubing, suggested by T. D. Noe, the 4th iteration yields an integer. Fractions are 4/3, 8/3, 64/3, followed by integers 9728, 920599396352, etc.

			a(2) = 8, the numerator of (4/3) * ceiling ((4/3)^2) = (4/3) * 2 = 8/3.
a(3) = 64, the numerator of (8/3) * ceiling ((8/3)^2) = (8/3) * 8 = 64/3.

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

Cf. A072340, A085276, A117596, A117620.

NestList[# Ceiling[#^2]&,4/3,6]//Numerator (* Harvey P. Dale, Mar 23 2019 *)

Data, comments, and examples corrected by Harvey P. Dale, Mar 23 2019

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A072340 Number of steps to reach an integer starting with n/3 and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A085285 Positions where records occur in A072340.

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A085286 Record values in A072340.

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A117596 Start with x=6/5; repeatedly apply the map x -> x*ceiling(x); sequence gives numerators of the resulting sequence of fractions.

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A117620 Start with x=4/3; repeatedly apply the map x -> (x^2) ceiling(x); sequence gives numerators of the resulting sequence of fractions.

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A117636 Start with x=4/3; repeatedly apply the map x -> x ceiling(x^2); sequence gives numerators of the resulting sequence of fractions.

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