A074097 -id:A074097 - cp's OEIS frontend

A074078 Number of steps to reach an integer starting with s = n/3 and iterating the map x -> s*ceiling(x).

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

Offset: 3

N. J. A. Sloane, Sep 16 2002

nonn

			s = 5/3 -> 10/3 -> 20/3 -> 35/3 -> 20, so a(5) = 4.

Michael S. Branicky, Table of n, a(n) for n = 3..10002

Cf. A072340, A074090, A074091, A074096. Records are in A074097, A074098.

First integer reached: A081852.

f := proc(b1,b2) local c1,c2,t1,t2,t3,t4,i; c1 := numer(b1/b2); c2 := denom(b1/b2); i := 0; while c2 <> 1 do i := i+1; t1 := ceil(c1/c2); t2 := b1*t1; t3 := t2/b2; c1 := numer(t3); c2 := denom(t3); od: RETURN(i); end; [seq(f(n,3),n=4..120)];

ce[n_] := Length[NestWhileList[(n/3)*Ceiling[#] &, n/3, ! IntegerQ[#] &]] - 1; Table[ce[n], {n, 3, 108}] (* Jayanta Basu, Jul 30 2013 *)

from math import ceil
from fractions import Fraction
def a(n):
    s = Fraction(n, 3)
    x, c = s, 0
    while x.denominator != 1:
        U = ceil(x)
        x, c = U*s, c+1
    return c
print([a(n) for n in range(3, 109)]) # Michael S. Branicky, Jan 09 2025

A074098 Where records occur in A074078.

Original entry on oeis.org

4, 5, 10, 275, 508, 3694, 5675, 6562, 32420, 137782, 295409, 430645, 13564046, 22289981, 387420490, 3486784402

Offset: 1

N. J. A. Sloane, Sep 17 2002

RECORDS transform of A074078.

N. J. A. Sloane, Transforms

Cf. A074078, A074097.

ce[n_] := Length[NestWhileList[(n/3)*Ceiling[#] &, n/3, ! IntegerQ[#] &]] - 1; t = {}; r = 0; Do[If[ce[n] > r, AppendTo[t, n]; r = ce[n]], {n, 3, 5*10^5}]; t (* Jayanta Basu, Jul 31 2013 *)

a(9)-a(12) from Jayanta Basu, Jul 31 2013
a(13)-a(14) from Sean A. Irvine, Jan 09 2025
a(15)-a(16) from Michael S. Branicky, Jan 09 2025

cp's OEIS Frontend

A074078 Number of steps to reach an integer starting with s = n/3 and iterating the map x -> s*ceiling(x).

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